Solving Non-linear Least Squares — Ceres Solver

# Solving Non-linear Least Squares

## Introduction

Effective use of Ceres Solver requires some familiarity with the basic components of a non-linear least squares solver, so before we describe how to configure and use the solver, we will take a brief look at how some of the core optimization algorithms in Ceres Solver work.

Let \(x \in \mathbb{R}^n\) be an \(n\)-dimensional vector of variables, and \(F(x) = \left[f_1(x), ... , f_{m}(x) \right]^{\top}\) be a \(m\)-dimensional function of \(x\). We are interested in solving the optimization problem [1]

(1)\[\begin{split}\arg \min_x \frac{1}{2}\|F(x)\|^2\ . \\ L \le x \le U\end{split}\]

Where, \(L\) and \(U\) are vector lower and upper bounds on the parameter vector \(x\). The inequality holds component-wise.

Since the efficient global minimization of (1) for general \(F(x)\) is an intractable problem, we will have to settle for finding a local minimum.

In the following, the Jacobian \(J(x)\) of \(F(x)\) is an \(m\times n\) matrix, where \(J_{ij}(x) = D_j f_i(x)\) and the gradient vector is \(g(x) = \nabla \frac{1}{2}\|F(x)\|^2 = J(x)^\top F(x)\).

The general strategy when solving non-linear optimization problems is to solve a sequence of approximations to the original problem [NocedalWright]. At each iteration, the approximation is solved to determine a correction \(\Delta x\) to the vector \(x\). For non-linear least squares, an approximation can be constructed by using the linearization \(F(x+\Delta x) \approx F(x) + J(x)\Delta x\), which leads to the following linear least squares problem:

(2)\[\min_{\Delta x} \frac{1}{2}\|J(x)\Delta x + F(x)\|^2\]

Unfortunately, naively solving a sequence of these problems and updating \(x \leftarrow x+ \Delta x\) leads to an algorithm that may not converge. To get a convergent algorithm, we need to control the size of the step \(\Delta x\). Depending on how the size of the step \(\Delta x\) is controlled, non-linear optimization algorithms can be divided into two major categories [NocedalWright].

1. Trust Region The trust region approach approximates the objective function using a model function (often a quadratic) over a subset of the search space known as the trust region. If the model function succeeds in minimizing the true objective function the trust region is expanded; conversely, otherwise it is contracted and the model optimization problem is solved again.

2. Line Search The line search approach first finds a descent direction along which the objective function will be reduced and then computes a step size that decides how far should move along that direction. The descent direction can be computed by various methods, such as gradient descent, Newton’s method and Quasi-Newton method. The step size can be determined either exactly or inexactly.

Trust region methods are in some sense dual to line search methods: trust region methods first choose a step size (the size of the trust region) and then a step direction while line search methods first choose a step direction and then a step size. Ceres Solver implements multiple algorithms in both categories.

## Trust Region Methods

The basic trust region algorithm looks something like this.

> 1. Given an initial point \(x\) and a trust region radius \(\mu\).

> 2. Solve

>    \[\begin{split}\arg \min_{\Delta x}& \frac{1}{2}\|J(x)\Delta x + F(x)\|^2 \\ \text{such that} &\|D(x)\Delta x\|^2 \le \mu\\ &L \le x + \Delta x \le U.\end{split}\]

> 3. \(\rho = \frac{\displaystyle \|F(x + \Delta x)\|^2 - \|F(x)\|^2}{\displaystyle \|J(x)\Delta x + F(x)\|^2 - \|F(x)\|^2}\)

> 4. if \(\rho > \epsilon\) then \(x = x + \Delta x\).

> 5. if \(\rho > \eta_1\) then \(\mu = 2 \mu\)

> 6. else if \(\rho < \eta_2\) then \(\mu = 0.5 * \mu\)

> 7. Go to 2.

Here, \(\mu\) is the trust region radius, \(D(x)\) is some matrix used to define a metric on the domain of \(F(x)\) and \(\rho\) measures the quality of the step \(\Delta x\), i.e., how well did the linear model predict the decrease in the value of the non-linear objective. The idea is to increase or decrease the radius of the trust region depending on how well the linearization predicts the behavior of the non-linear objective, which in turn is reflected in the value of \(\rho\).

The key computational step in a trust-region algorithm is the solution of the constrained optimization problem

(3)\[\begin{split}\arg \min_{\Delta x}&\quad \frac{1}{2}\|J(x)\Delta x + F(x)\|^2 \\ \text{such that} &\quad \|D(x)\Delta x\|^2 \le \mu\\ &\quad L \le x + \Delta x \le U.\end{split}\]

There are a number of different ways of solving this problem, each giving rise to a different concrete trust-region algorithm. Currently, Ceres implements two trust-region algorithms - Levenberg-Marquardt and Dogleg, each of which is augmented with a line search if bounds constraints are present [Kanzow]. The user can choose between them by setting `Solver::Options::trust_region_strategy_type`.

Footnotes

[1]

At the level of the non-linear solver, the block structure is not relevant, therefore our discussion here is in terms of an optimization problem defined over a state vector of size \(n\). Similarly the presence of loss functions is also ignored as the problem is internally converted into a pure non-linear least squares problem.

### Levenberg-Marquardt

The Levenberg-Marquardt algorithm [Levenberg] [Marquardt] is the most popular algorithm for solving non-linear least squares problems. It was also the first trust region algorithm to be developed [Levenberg] [Marquardt]. Ceres implements an exact step [Madsen] and an inexact step variant of the Levenberg-Marquardt algorithm [WrightHolt] [NashSofer].

It can be shown, that the solution to (3) can be obtained by solving an unconstrained optimization of the form

(4)\[\arg\min_{\Delta x} \frac{1}{2}\|J(x)\Delta x + F(x)\|^2 +\lambda \|D(x)\Delta x\|^2\]

Where, \(\lambda\) is a Lagrange multiplier that is inversely related to \(\mu\). In Ceres, we solve for

(5)\[\arg\min_{\Delta x} \frac{1}{2}\|J(x)\Delta x + F(x)\|^2 + \frac{1}{\mu} \|D(x)\Delta x\|^2\]

The matrix \(D(x)\) is a non-negative diagonal matrix, typically the square root of the diagonal of the matrix \(J(x)^\top J(x)\).

Before going further, let us make some notational simplifications.

We will assume that the matrix \(\frac{1}{\sqrt{\mu}} D\) has been concatenated at the bottom of the matrix \(J(x)\) and a corresponding vector of zeroes has been added to the bottom of \(F(x)\), i.e.:

\[\begin{split}J(x) = \begin{bmatrix} J(x) \\ \frac{1}{\sqrt{\mu}} D \end{bmatrix},\quad F(x) = \begin{bmatrix} F(x) \\ 0 \end{bmatrix}.\end{split}\]

This allows us to re-write (5) as

(6)\[\min_{\Delta x} \frac{1}{2} \|J(x)\Delta x + F(x)\|^2 .\]

and only talk about \(J(x)\) and \(F(x)\) going forward.

For all but the smallest problems the solution of (6) in each iteration of the Levenberg-Marquardt algorithm is the dominant computational cost. Ceres provides a number of different options for solving (6). There are two major classes of methods - factorization and iterative.

The factorization methods are based on computing an exact solution of (5) using a Cholesky or a QR factorization and lead to the so called exact step Levenberg-Marquardt algorithm. But it is not clear if an exact solution of (5) is necessary at each step of the Levenberg-Mardquardt algorithm. We have already seen evidence that this may not be the case, as (5) is itself a regularized version of (2). Indeed, it is possible to construct non-linear optimization algorithms in which the linearized problem is solved approximately. These algorithms are known as inexact Newton or truncated Newton methods [NocedalWright].

An inexact Newton method requires two ingredients. First, a cheap method for approximately solving systems of linear equations. Typically an iterative linear solver like the Conjugate Gradients method is used for this purpose [NocedalWright]. Second, a termination rule for the iterative solver. A typical termination rule is of the form

(7)\[\|H(x) \Delta x + g(x)\| \leq \eta_k \|g(x)\|.\]

Here, \(k\) indicates the Levenberg-Marquardt iteration number and \(0 < \eta_k <1\) is known as the forcing sequence. [WrightHolt] prove that a truncated Levenberg-Marquardt algorithm that uses an inexact Newton step based on (7) converges for any sequence \(\eta_k \leq \eta_0 < 1\) and the rate of convergence depends on the choice of the forcing sequence \(\eta_k\).

Ceres supports both exact and inexact step solution strategies. When the user chooses a factorization based linear solver, the exact step Levenberg-Marquardt algorithm is used. When the user chooses an iterative linear solver, the inexact step Levenberg-Marquardt algorithm is used.

We will talk more about the various linear solvers that you can use in Linear Solvers.

### Dogleg

Another strategy for solving the trust region problem (3) was introduced by M. J. D. Powell. The key idea there is to compute two vectors

\[\begin{split}\Delta x^{\text{Gauss-Newton}} &= \arg \min_{\Delta x}\frac{1}{2} \|J(x)\Delta x + f(x)\|^2.\\ \Delta x^{\text{Cauchy}} &= -\frac{\|g(x)\|^2}{\|J(x)g(x)\|^2}g(x).\end{split}\]

Note that the vector \(\Delta x^{\text{Gauss-Newton}}\) is the solution to (2) and \(\Delta x^{\text{Cauchy}}\) is the vector that minimizes the linear approximation if we restrict ourselves to moving along the direction of the gradient. Dogleg methods finds a vector \(\Delta x\) defined by \(\Delta x^{\text{Gauss-Newton}}\) and \(\Delta x^{\text{Cauchy}}\) that solves the trust region problem. Ceres supports two variants that can be chose by setting `Solver::Options::dogleg_type`.

`TRADITIONAL_DOGLEG` as described by Powell, constructs two line segments using the Gauss-Newton and Cauchy vectors and finds the point farthest along this line shaped like a dogleg (hence the name) that is contained in the trust-region. For more details on the exact reasoning and computations, please see Madsen et al [Madsen].

`SUBSPACE_DOGLEG` is a more sophisticated method that considers the entire two dimensional subspace spanned by these two vectors and finds the point that minimizes the trust region problem in this subspace [ByrdSchnabel].

The key advantage of the Dogleg over Levenberg-Marquardt is that if the step computation for a particular choice of \(\mu\) does not result in sufficient decrease in the value of the objective function, Levenberg-Marquardt solves the linear approximation from scratch with a smaller value of \(\mu\). Dogleg on the other hand, only needs to compute the interpolation between the Gauss-Newton and the Cauchy vectors, as neither of them depend on the value of \(\mu\). As a result the Dogleg method only solves one linear system per successful step, while Levenberg-Marquardt may need to solve an arbitrary number of linear systems before it can make progress [LourakisArgyros].

A disadvantage of the Dogleg implementation in Ceres Solver is that is can only be used with method can only be used with exact factorization based linear solvers.

### Inner Iterations

Some non-linear least squares problems have additional structure in the way the parameter blocks interact that it is beneficial to modify the way the trust region step is computed. For example, consider the following regression problem

\[y = a_1 e^{b_1 x} + a_2 e^{b_3 x^2 + c_1}\]

Given a set of pairs \(\{(x_i, y_i)\}\), the user wishes to estimate \(a_1, a_2, b_1, b_2\), and \(c_1\).

Notice that the expression on the left is linear in \(a_1\) and \(a_2\), and given any value for \(b_1, b_2\) and \(c_1\), it is possible to use linear regression to estimate the optimal values of \(a_1\) and \(a_2\). It’s possible to analytically eliminate the variables \(a_1\) and \(a_2\) from the problem entirely. Problems like these are known as separable least squares problem and the most famous algorithm for solving them is the Variable Projection algorithm invented by Golub & Pereyra [GolubPereyra].

Similar structure can be found in the matrix factorization with missing data problem. There the corresponding algorithm is known as Wiberg’s algorithm [Wiberg].

Ruhe & Wedin present an analysis of various algorithms for solving separable non-linear least squares problems and refer to Variable Projection as Algorithm I in their paper [RuheWedin].

Implementing Variable Projection is tedious and expensive. Ruhe & Wedin present a simpler algorithm with comparable convergence properties, which they call Algorithm II. Algorithm II performs an additional optimization step to estimate \(a_1\) and \(a_2\) exactly after computing a successful Newton step.

This idea can be generalized to cases where the residual is not linear in \(a_1\) and \(a_2\), i.e.,

\[y = f_1(a_1, e^{b_1 x}) + f_2(a_2, e^{b_3 x^2 + c_1})\]

In this case, we solve for the trust region step for the full problem, and then use it as the starting point to further optimize just a_1 and a_2. For the linear case, this amounts to doing a single linear least squares solve. For non-linear problems, any method for solving the \(a_1\) and \(a_2\) optimization problems will do. The only constraint on \(a_1\) and \(a_2\) (if they are two different parameter block) is that they do not co-occur in a residual block.

This idea can be further generalized, by not just optimizing \((a_1, a_2)\), but decomposing the graph corresponding to the Hessian matrix’s sparsity structure into a collection of non-overlapping independent sets and optimizing each of them.

Setting `Solver::Options::use_inner_iterations` to `true` enables the use of this non-linear generalization of Ruhe & Wedin’s Algorithm II. This version of Ceres has a higher iteration complexity, but also displays better convergence behavior per iteration.

Setting `Solver::Options::num_threads` to the maximum number possible is highly recommended.

### Non-monotonic Steps

Note that the basic trust-region algorithm described in Trust Region Methods is a descent algorithm in that it only accepts a point if it strictly reduces the value of the objective function.

Relaxing this requirement allows the algorithm to be more efficient in the long term at the cost of some local increase in the value of the objective function.

This is because allowing for non-decreasing objective function values in a principled manner allows the algorithm to jump over boulders as the method is not restricted to move into narrow valleys while preserving its convergence properties.

Setting `Solver::Options::use_nonmonotonic_steps` to `true` enables the non-monotonic trust region algorithm as described by Conn, Gould & Toint in [Conn].

Even though the value of the objective function may be larger than the minimum value encountered over the course of the optimization, the final parameters returned to the user are the ones corresponding to the minimum cost over all iterations.

The option to take non-monotonic steps is available for all trust region strategies.

## Line Search Methods

Note

The line search method in Ceres Solver cannot handle bounds constraints right now, so it can only be used for solving unconstrained problems.

The basic line search algorithm looks something like this:

> 1. Given an initial point \(x\)

> 2. \(\Delta x = -H^{-1}(x) g(x)\)

> 3. \(\arg \min_\mu \frac{1}{2} \| F(x + \mu \Delta x) \|^2\)

> 4. \(x = x + \mu \Delta x\)

> 5. Goto 2.

Here \(H(x)\) is some approximation to the Hessian of the objective function, and \(g(x)\) is the gradient at \(x\). Depending on the choice of \(H(x)\) we get a variety of different search directions \(\Delta x\).

Step 4, which is a one dimensional optimization or Line Search along \(\Delta x\) is what gives this class of methods its name.

Different line search algorithms differ in their choice of the search direction \(\Delta x\) and the method used for one dimensional optimization along \(\Delta x\). The choice of \(H(x)\) is the primary source of computational complexity in these methods. Currently, Ceres Solver supports four choices of search directions, all aimed at large scale problems.

1. `STEEPEST_DESCENT` This corresponds to choosing \(H(x)\) to be the identity matrix. This is not a good search direction for anything but the simplest of the problems. It is only included here for completeness.

2. `NONLINEAR_CONJUGATE_GRADIENT` A generalization of the Conjugate Gradient method to non-linear functions. The generalization can be performed in a number of different ways, resulting in a variety of search directions. Ceres Solver currently supports `FLETCHER_REEVES`, `POLAK_RIBIERE` and `HESTENES_STIEFEL` directions.

3. `BFGS` A generalization of the Secant method to multiple dimensions in which a full, dense approximation to the inverse Hessian is maintained and used to compute a quasi-Newton step [NocedalWright]. `BFGS` and its limited memory variant `LBFGS` are currently the best known general quasi-Newton algorithm.

4. `LBFGS` A limited memory approximation to the full `BFGS` method in which the last M iterations are used to approximate the inverse Hessian used to compute a quasi-Newton step [Nocedal], [ByrdNocedal].

Currently Ceres Solver supports both a backtracking and interpolation based Armijo line search algorithm (`ARMIJO`) , and a sectioning / zoom interpolation (strong) Wolfe condition line search algorithm (`WOLFE`).

Note

In order for the assumptions underlying the `BFGS` and `LBFGS` methods to be satisfied the `WOLFE` algorithm must be used.

## Linear Solvers

Observe that for both of the trust-region methods described above, the key computational cost is the solution of a linear least squares problem of the form

(8)\[\min_{\Delta x} \frac{1}{2} \|J(x)\Delta x + F(x)\|^2 .\]

Let \(H(x)= J(x)^\top J(x)\) and \(g(x) = -J(x)^\top F(x)\). For notational convenience let us also drop the dependence on \(x\). Then it is easy to see that solving (8) is equivalent to solving the normal equations.

(9)\[H \Delta x = g\]

Ceres provides a number of different options for solving (9).

### DENSE_QR

For small problems (a couple of hundred parameters and a few thousand residuals) with relatively dense Jacobians, QR-decomposition is the method of choice [Bjorck]. Let \(J = QR\) be the QR-decomposition of \(J\), where \(Q\) is an orthonormal matrix and \(R\) is an upper triangular matrix [TrefethenBau]. Then it can be shown that the solution to (9) is given by

\[\Delta x^* = -R^{-1}Q^\top f\]

You can use QR-decomposition by setting `Solver::Options::linear_solver_type` to `DENSE_QR`.

By default (`Solver::Options::dense_linear_algebra_library_type = EIGEN`) Ceres Solver will use Eigen Householder QR factorization .

If Ceres Solver has been built with an optimized LAPACK implementation, then the user can also choose to use LAPACK’s DGEQRF routine by setting `Solver::Options::dense_linear_algebra_library_type` to `LAPACK`. Depending on the LAPACK and the underlying BLAS implementation this may perform better than using Eigen’s Householder QR factorization.

If an NVIDIA GPU is available and Ceres Solver has been built with CUDA support enabled, then the user can also choose to perform the QR-decomposition on the GPU by setting `Solver::Options::dense_linear_algebra_library_type` to `CUDA`. Depending on the GPU this can lead to a substantial speedup. Using CUDA only makes sense for moderate to large sized problems. This is because to perform the decomposition on the GPU the matrix \(J\) needs to be transferred from the CPU to the GPU and this incurs a cost. So unless the speedup from doing the decomposition on the GPU is large enough to also account for the time taken to transfer the Jacobian to the GPU, using CUDA will not be better than just doing the decomposition on the CPU.

### DENSE_NORMAL_CHOLESKY

It is often the case that the number of rows in the Jacobian \(J\) are much larger than the the number of columns. The complexity of QR factorization scales linearly with the number of rows, so beyond a certain size it is more efficient to solve (9) using a dense Cholesky factorization.

Let \(H = R^\top R\) be the Cholesky factorization of the normal equations, where \(R\) is an upper triangular matrix, then the solution to (9) is given by

\[\Delta x^* = R^{-1} R^{-\top} g.\]

The observant reader will note that the \(R\) in the Cholesky factorization of \(H\) is the same upper triangular matrix \(R\) in the QR factorization of \(J\). Since \(Q\) is an orthonormal matrix, \(J=QR\) implies that \(J^\top J = R^\top Q^\top Q R = R^\top R\).

Unfortunately, forming the matrix \(H = J'J\) squares the condition number. As a result while the cost of forming \(H\) and computing its Cholesky factorization is lower than computing the QR-factorization of \(J\), we pay the price in terms of increased numerical instability and potential failure of the Cholesky factorization for ill-conditioned Jacobians.

You can use dense Cholesky factorization by setting `Solver::Options::linear_solver_type` to `DENSE_NORMAL_CHOLESKY`.

By default (`Solver::Options::dense_linear_algebra_library_type = EIGEN`) Ceres Solver will use Eigen’s LLT factorization routine.

If Ceres Solver has been built with an optimized LAPACK implementation, then the user can also choose to use LAPACK’s DPOTRF routine by setting `Solver::Options::dense_linear_algebra_library_type` to `LAPACK`. Depending on the LAPACK and the underlying BLAS implementation this may perform better than using Eigen’s Cholesky factorization.

If an NVIDIA GPU is available and Ceres Solver has been built with CUDA support enabled, then the user can also choose to perform the Cholesky factorization on the GPU by setting `Solver::Options::dense_linear_algebra_library_type` to `CUDA`. Depending on the GPU this can lead to a substantial speedup. Using CUDA only makes sense for moderate to large sized problems. This is because to perform the decomposition on the GPU the matrix \(H\) needs to be transferred from the CPU to the GPU and this incurs a cost. So unless the speedup from doing the decomposition on the GPU is large enough to also account for the time taken to transfer the Jacobian to the GPU, using CUDA will not be better than just doing the decomposition on the CPU.

### SPARSE_NORMAL_CHOLESKY

Large non-linear least square problems are usually sparse. In such cases, using a dense QR or Cholesky factorization is inefficient. For such problems, Cholesky factorization routines which treat \(H\) as a sparse matrix and computes a sparse factor \(R\) are better suited [Davis]. This can lead to substantial savings in memory and CPU time for large sparse problems.

You can use dense Cholesky factorization by setting `Solver::Options::linear_solver_type` to `SPARSE_NORMAL_CHOLESKY`.

The use of this linear solver requires that Ceres is compiled with support for at least one of:

> 1. SuiteSparse (`SUITE_SPARSE`).

> 2. Apple’s Accelerate framework (`ACCELERATE_SPARSE`).

> 3. Eigen’s sparse linear solvers (`EIGEN_SPARSE`).

SuiteSparse and Accelerate offer high performance sparse Cholesky factorization routines as they level-3 BLAS routines internally. Eigen’s sparse Cholesky routines are simplicial and do not use dense linear algebra routines and as a result cannot compete with SuiteSparse and Accelerate, especially on large problems. As a result to get the best performance out of SuiteSparse it should be linked to high quality BLAS and LAPACK implementations e.g. ATLAS, OpenBLAS or Intel MKL.

A critical part of a sparse Cholesky factorization routine is the use a fill-reducing ordering. By default Ceres Solver uses the Approximate Minimum Degree (`AMD`) ordering, which usually performs well, but there are other options that may perform better depending on the actual sparsity structure of the Jacobian. See Ordering for more details.

### CGNR

For general sparse problems, if the problem is too large for sparse Cholesky factorization or a sparse linear algebra library is not linked into Ceres, another option is the `CGNR` solver. This solver uses the Conjugate Gradients <https://en.wikipedia.org/wiki/Conjugate_gradient_method>_ method on the normal equations, but without forming the normal equations explicitly. It exploits the relation

\[H x = J^\top J x = J^\top(J x)\]

Because `CGNR` never solves the linear system exactly, when the user chooses `CGNR` as the linear solver, Ceres automatically switches from the exact step algorithm to an inexact step algorithm. This also means that `CGNR` can only be used with `LEVENBERG_MARQUARDT` and not with `DOGLEG` trust region strategy.

`CGNR` by default runs on the CPU. However, if an NVIDIA GPU is available and Ceres Solver has been built with CUDA support enabled, then the user can also choose to run `CGNR` on the GPU by setting `Solver::Options::sparse_linear_algebra_library_type` to `CUDA_SPARSE`. The key complexity of `CGNR` comes from evaluating the two sparse-matrix vector products (SpMV) \(Jx\) and \(J'y\). GPUs are particularly well suited for doing sparse matrix-vector products. As a result, for large problems using a GPU can lead to a substantial speedup.

The convergence of Conjugate Gradients depends on the conditioner number \(\kappa(H)\). Usually \(H\) is quite poorly conditioned and a Preconditioner must be used to get reasonable performance. See section on Preconditioners for more details.

### DENSE_SCHUR & SPARSE_SCHUR

While it is possible to use `SPARSE_NORMAL_CHOLESKY` to solve bundle adjustment problems, they have a special sparsity structure that can be exploited to solve the normal equations more efficiently.

Suppose that the bundle adjustment problem consists of \(p\) cameras and \(q\) points and the variable vector \(x\) has the block structure \(x = [y_{1}, ... ,y_{p},z_{1}, ... ,z_{q}]\). Where, \(y\) and \(z\) correspond to camera and point parameters respectively. Further, let the camera blocks be of size \(c\) and the point blocks be of size \(s\) (for most problems \(c\) = \(6\)–9 and \(s = 3\)). Ceres does not impose any constancy requirement on these block sizes, but choosing them to be constant simplifies the exposition.

The key property of bundle adjustment problems which we will exploit is the fact that no term \(f_{i}\) in (1) includes two or more point blocks at the same time. This in turn implies that the matrix \(H\) is of the form

(10)\[\begin{split}H = \left[ \begin{matrix} B & E\\ E^\top & C \end{matrix} \right]\ ,\end{split}\]

where \(B \in \mathbb{R}^{pc\times pc}\) is a block sparse matrix with \(p\) blocks of size \(c\times c\) and \(C \in \mathbb{R}^{qs\times qs}\) is a block diagonal matrix with \(q\) blocks of size \(s\times s\). \(E \in \mathbb{R}^{pc\times qs}\) is a general block sparse matrix, with a block of size \(c\times s\) for each observation. Let us now block partition \(\Delta x = [\Delta y,\Delta z]\) and \(g=[v,w]\) to restate (9) as the block structured linear system

(11)\[\begin{split}\left[ \begin{matrix} B & E\\ E^\top & C \end{matrix} \right]\left[ \begin{matrix} \Delta y \\ \Delta z \end{matrix} \right] = \left[ \begin{matrix} v\\ w \end{matrix} \right]\ ,\end{split}\]

and apply Gaussian elimination to it. As we noted above, \(C\) is a block diagonal matrix, with small diagonal blocks of size \(s\times s\). Thus, calculating the inverse of \(C\) by inverting each of these blocks is cheap. This allows us to eliminate \(\Delta z\) by observing that \(\Delta z = C^{-1}(w - E^\top \Delta y)\), giving us

(12)\[\left[B - EC^{-1}E^\top\right] \Delta y = v - EC^{-1}w\ .\]

The matrix

\[S = B - EC^{-1}E^\top\]

is the Schur complement of \(C\) in \(H\). It is also known as the reduced camera matrix, because the only variables participating in (12) are the ones corresponding to the cameras. \(S \in \mathbb{R}^{pc\times pc}\) is a block structured symmetric positive definite matrix, with blocks of size \(c\times c\). The block \(S_{ij}\) corresponding to the pair of images \(i\) and \(j\) is non-zero if and only if the two images observe at least one common point.

Now (11) can be solved by first forming \(S\), solving for \(\Delta y\), and then back-substituting \(\Delta y\) to obtain the value of \(\Delta z\). Thus, the solution of what was an \(n\times n\), \(n=pc+qs\) linear system is reduced to the inversion of the block diagonal matrix \(C\), a few matrix-matrix and matrix-vector multiplies, and the solution of block sparse \(pc\times pc\) linear system (12). For almost all problems, the number of cameras is much smaller than the number of points, \(p \ll q\), thus solving (12) is significantly cheaper than solving (11). This is the Schur complement trick [Brown].

This still leaves open the question of solving (12). As we discussed when considering the exact solution of the normal equations using Cholesky factorization, we have two options.

1. `DENSE_SCHUR` - The first is dense Cholesky factorization, where we store and factor \(S\) as a dense matrix. This method has \(O(p^2)\) space complexity and \(O(p^3)\) time complexity and is only practical for problems with up to a few hundred cameras.

2. `SPARSE_SCHUR` - For large bundle adjustment problems \(S\) is typically a fairly sparse matrix, as most images only see a small fraction of the scene. This leads us to the second option: sparse Cholesky factorization [Davis]. Here we store \(S\) as a sparse matrix, use row and column re-ordering algorithms to maximize the sparsity of the Cholesky decomposition, and focus their compute effort on the non-zero part of the factorization [Davis] [Chen] . Sparse direct methods, depending on the exact sparsity structure of the Schur complement, allow bundle adjustment algorithms to scenes with thousands of cameras.

### ITERATIVE_SCHUR

Another option for bundle adjustment problems is to apply Conjugate Gradients to the reduced camera matrix \(S\) instead of \(H\). One reason to do this is that \(S\) is a much smaller matrix than \(H\), but more importantly, it can be shown that \(\kappa(S)\leq \kappa(H)\) [Agarwal].

Ceres implements Conjugate Gradients on \(S\) as the `ITERATIVE_SCHUR` solver. When the user chooses `ITERATIVE_SCHUR` as the linear solver, Ceres automatically switches from the exact step algorithm to an inexact step algorithm.

The key computational operation when using Conjuagate Gradients is the evaluation of the matrix vector product \(Sx\) for an arbitrary vector \(x\). Because PCG only needs access to \(S\) via its product with a vector, one way to evaluate \(Sx\) is to observe that

(13)\[\begin{split}x_1 &= E^\top x\\ x_2 &= C^{-1} x_1\\ x_3 &= Ex_2\\ x_4 &= Bx\\ Sx &= x_4 - x_3\end{split}\]

Thus, we can run Conjugate Gradients on \(S\) with the same computational effort per iteration as Conjugate Gradients on \(H\), while reaping the benefits of a more powerful preconditioner. In fact, we do not even need to compute \(H\), (13) can be implemented using just the columns of \(J\).

Equation (13) is closely related to Domain Decomposition methods for solving large linear systems that arise in structural engineering and partial differential equations. In the language of Domain Decomposition, each point in a bundle adjustment problem is a domain, and the cameras form the interface between these domains. The iterative solution of the Schur complement then falls within the sub-category of techniques known as Iterative Sub-structuring [Saad] [Mathew].

While in most cases the above method for evaluating \(Sx\) is the way to go, for some problems it is better to compute the Schur complemenent \(S\) explicitly and then run Conjugate Gradients on it. This can be done by settin `Solver::Options::use_explicit_schur_complement` to `true`. This option can only be used with the `SCHUR_JACOBI` preconditioner.

### SCHUR_POWER_SERIES_EXPANSION

It can be shown that the inverse of the Schur complement can be written as an infinite power-series [Weber] [Zheng]:

\[\begin{split}S &= B - EC^{-1}E^\top\\ &= B(I - B^{-1}EC^{-1}E^\top)\\ S^{-1} &= (I - B^{-1}EC^{-1}E^\top)^{-1} B^{-1}\\ & = \sum_{i=0}^\infty \left(B^{-1}EC^{-1}E^\top\right)^{i} B^{-1}\end{split}\]

As a result a truncated version of this power series expansion can be used to approximate the inverse and therefore the solution to (12). Ceres allows the user to use Schur power series expansion in three ways.

1. As a linear solver. This is what [Weber] calls Power Bundle Adjustment and corresponds to using the truncated power series to approximate the inverse of the Schur complement. This is done by setting the following options.

       Solver::Options::linear_solver_type = ITERATIVE_SCHUR
       Solver::Options::preconditioner_type = IDENTITY
       Solver::Options::use_spse_initialization = true
       Solver::Options::max_linear_solver_iterations = 0;

       // The following two settings are worth tuning for your application.
       Solver::Options::max_num_spse_iterations = 5;
       Solver::Options::spse_tolerance = 0.1;

2. As a preconditioner for `ITERATIVE_SCHUR`. Any method for approximating the inverse of a matrix can also be used as a preconditioner. This is enabled by setting the following options.

       Solver::Options::linear_solver_type = ITERATIVE_SCHUR
       Solver::Options::preconditioner_type = SCHUR_POWER_SERIES_EXPANSION;
       Solver::Options::use_spse_initialization = false;

       // This is worth tuning for your application.
       Solver::Options::max_num_spse_iterations = 5;

3. As initialization for `ITERATIIVE_SCHUR` with any preconditioner. This is a combination of the above two, where the Schur Power Series Expansion

       Solver::Options::linear_solver_type = ITERATIVE_SCHUR
       Solver::Options::preconditioner_type = ... // Preconditioner of your choice.
       Solver::Options::use_spse_initialization = true
       Solver::Options::max_linear_solver_iterations = 0;

       // The following two settings are worth tuning for your application.
       Solver::Options::max_num_spse_iterations = 5;
       // This only affects the initialization but not the preconditioner.
       Solver::Options::spse_tolerance = 0.1;

## Mixed Precision Solves

Generally speaking Ceres Solver does all its arithmetic in double precision. Sometimes though, one can use single precision arithmetic to get substantial speedups. Currently, for linear solvers that perform Cholesky factorization (sparse or dense) the user has the option cast the linear system to single precision and then use single precision Cholesky factorization routines to solve the resulting linear system. This can be enabled by setting `Solver::Options::use_mixed_precision_solves` to `true`.

Depending on the conditioning of the problem, the use of single precision factorization may lead to some loss of accuracy. Some of this accuracy can be recovered by performing Iterative Refinement. The number of iterations of iterative refinement are controlled by `Solver::Options::max_num_refinement_iterations`. The default value of this parameter is zero, which means if `Solver::Options::use_mixed_precision_solves` is `true`, then no iterative refinement is performed. Usually 2-3 refinement iterations are enough.

Mixed precision solves are available in the following linear solver configurations:

1. `DENSE_NORMAL_CHOLESKY` + `EIGEN`/ `LAPACK` / `CUDA`.

2. `DENSE_SCHUR` + `EIGEN`/ `LAPACK` / `CUDA`.

3. `SPARSE_NORMAL_CHOLESKY` + `EIGEN_SPARSE` / `ACCELERATE_SPARSE`

4. `SPARSE_SCHUR` + `EIGEN_SPARSE` / `ACCELERATE_SPARSE`

Mixed precision solves area not available when using `SUITE_SPARSE` as the sparse linear algebra backend because SuiteSparse/CHOLMOD does not support single precision solves.

## Preconditioners

The convergence rate of Conjugate Gradients for solving (9) depends on the distribution of eigenvalues of \(H\) [Saad]. A useful upper bound is \(\sqrt{\kappa(H)}\), where, \(\kappa(H)\) is the condition number of the matrix \(H\). For most non-linear least squares problems, \(\kappa(H)\) is high and a direct application of Conjugate Gradients to (9) results in extremely poor performance.

The solution to this problem is to replace (9) with a preconditioned system. Given a linear system, \(Ax =b\) and a preconditioner \(M\) the preconditioned system is given by \(M^{-1}Ax = M^{-1}b\). The resulting algorithm is known as Preconditioned Conjugate Gradients algorithm (PCG) and its worst case complexity now depends on the condition number of the preconditioned matrix \(\kappa(M^{-1}A)\).

The computational cost of using a preconditioner \(M\) is the cost of computing \(M\) and evaluating the product \(M^{-1}y\) for arbitrary vectors \(y\). Thus, there are two competing factors to consider: How much of \(H\)’s structure is captured by \(M\) so that the condition number \(\kappa(HM^{-1})\) is low, and the computational cost of constructing and using \(M\). The ideal preconditioner would be one for which \(\kappa(M^{-1}A) =1\). \(M=A\) achieves this, but it is not a practical choice, as applying this preconditioner would require solving a linear system equivalent to the unpreconditioned problem. It is usually the case that the more information \(M\) has about \(H\), the more expensive it is use. For example, Incomplete Cholesky factorization based preconditioners have much better convergence behavior than the Jacobi preconditioner, but are also much more expensive.

For a survey of the state of the art in preconditioning linear least squares problems with general sparsity structure see [GouldScott].

Ceres Solver comes with an number of preconditioners suited for problems with general sparsity as well as the special sparsity structure encountered in bundle adjustment problems.

### IDENTITY

This is equivalent to using an identity matrix as a preconditioner, i.e. no preconditioner at all.

### JACOBI

The simplest of all preconditioners is the diagonal or Jacobi preconditioner, i.e., \(M=\operatorname{diag}(A)\), which for block structured matrices like \(H\) can be generalized to the block Jacobi preconditioner. The `JACOBI` preconditioner in Ceres when used with CGNR refers to the block diagonal of \(H\) and when used with ITERATIVE_SCHUR refers to the block diagonal of \(B\) [Mandel].

For detailed performance data about the performance of `JACOBI` on bundle adjustment problems see [Agarwal].

### SCHUR_JACOBI

Another obvious choice for ITERATIVE_SCHUR is the block diagonal of the Schur complement matrix \(S\), i.e, the block Jacobi preconditioner for \(S\). In Ceres we refer to it as the `SCHUR_JACOBI` preconditioner.

For detailed performance data about the performance of `SCHUR_JACOBI` on bundle adjustment problems see [Agarwal].

### CLUSTER_JACOBI and CLUSTER_TRIDIAGONAL

For bundle adjustment problems arising in reconstruction from community photo collections, more effective preconditioners can be constructed by analyzing and exploiting the camera-point visibility structure of the scene.

The key idea is to cluster the cameras based on the visibility structure of the scene. The similarity between a pair of cameras \(i\) and \(j\) is given by:

> \[S_{ij} = \frac{|V_i \cap V_j|}{|V_i| |V_j|}\]

Here \(V_i\) is the set of scene points visible in camera \(i\). This idea was first exploited by [KushalAgarwal] to create the `CLUSTER_JACOBI` and the `CLUSTER_TRIDIAGONAL` preconditioners which Ceres implements.

The performance of these two preconditioners depends on the speed and clustering quality of the clustering algorithm used when building the preconditioner. In the original paper, [KushalAgarwal] used the Canonical Views algorithm [Simon], which while producing high quality clusterings can be quite expensive for large graphs. So, Ceres supports two visibility clustering algorithms - `CANONICAL_VIEWS` and `SINGLE_LINKAGE`. The former is as the name implies Canonical Views algorithm of [Simon]. The latter is the the classic Single Linkage Clustering algorithm. The choice of clustering algorithm is controlled by `Solver::Options::visibility_clustering_type`.

### SCHUR_POWER_SERIES_EXPANSION

As explained in SCHUR_POWER_SERIES_EXPANSION, the Schur complement matrix admits a power series expansion and a truncated version of this power series can be used as a preconditioner for `ITERATIVE_SCHUR`. When used as a preconditioner `Solver::Options::max_num_spse_iterations` controls the number of terms in the power series that are used.

### SUBSET

This is a preconditioner for problems with general sparsity. Given a subset of residual blocks of a problem, it uses the corresponding subset of the rows of the Jacobian to construct a preconditioner [Dellaert].

Suppose the Jacobian \(J\) has been horizontally partitioned as

> \[\begin{split}J = \begin{bmatrix} P \\ Q \end{bmatrix}\end{split}\]

Where, \(Q\) is the set of rows corresponding to the residual blocks in `Solver::Options::residual_blocks_for_subset_preconditioner`. The preconditioner is the matrix \((Q^\top Q)^{-1}\).

The efficacy of the preconditioner depends on how well the matrix \(Q\) approximates \(J^\top J\), or how well the chosen residual blocks approximate the full problem.

This preconditioner is NOT available when running `CGNR` using `CUDA`.

## Ordering

The order in which variables are eliminated in a linear solver can have a significant of impact on the efficiency and accuracy of the method. For example when doing sparse Cholesky factorization, there are matrices for which a good ordering will give a Cholesky factor with \(O(n)\) storage, whereas a bad ordering will result in an completely dense factor.

Ceres allows the user to provide varying amounts of hints to the solver about the variable elimination ordering to use. This can range from no hints, where the solver is free to decide the best possible ordering based on the user’s choices like the linear solver being used, to an exact order in which the variables should be eliminated, and a variety of possibilities in between.

The simplest thing to do is to just set `Solver::Options::linear_solver_ordering_type` to `AMD` (default) or `NESDIS` based on your understanding of the problem or empirical testing.

More information can be commmuniucated by using an instance `ParameterBlockOrdering` class.

Formally an ordering is an ordered partitioning of the parameter blocks, i.e, each parameter block belongs to exactly one group, and each group has a unique non-negative integer associated with it, that determines its order in the set of groups.

e.g. Consider the linear system

\[\begin{split}x + y &= 3 \\ 2x + 3y &= 7\end{split}\]

There are two ways in which it can be solved. First eliminating \(x\) from the two equations, solving for \(y\) and then back substituting for \(x\), or first eliminating \(y\), solving for \(x\) and back substituting for \(y\). The user can construct three orderings here.

1. \(\{0: x\}, \{1: y\}\) - eliminate \(x\) first.

2. \(\{0: y\}, \{1: x\}\) - eliminate \(y\) first.

3. \(\{0: x, y\}\) - Solver gets to decide the elimination order.

Thus, to have Ceres determine the ordering automatically, put all the variables in group 0 and to control the ordering for every variable, create groups \(0 \dots N-1\), one per variable, in the desired order.

`linear_solver_ordering == nullptr` and an ordering where all the parameter blocks are in one elimination group mean the same thing - the solver is free to choose what it thinks is the best elimination ordering using the ordering algorithm (specified using `Solver::Options::linear_solver_ordering_type`). Therefore in the following we will only consider the case where `linear_solver_ordering != nullptr`.

The exact interpretation of the `linear_solver_ordering` depends on the values of `Solver::Options::linear_solver_ordering_type`, `Solver::Options::linear_solver_type`, `Solver::Options::preconditioner_type` and `Solver::Options::sparse_linear_algebra_library_type` as we will explain below.

### Bundle Adjustment

If the user is using one of the Schur solvers (`DENSE_SCHUR`, `SPARSE_SCHUR`, `ITERATIVE_SCHUR`) and chooses to specify an ordering, it must have one important property. The lowest numbered elimination group must form an independent set in the graph corresponding to the Hessian, or in other words, no two parameter blocks in the first elimination group should co-occur in the same residual block. For the best performance, this elimination group should be as large as possible. For standard bundle adjustment problems, this corresponds to the first elimination group containing all the 3d points, and the second containing the parameter blocks for all the cameras.

If the user leaves the choice to Ceres, then the solver uses an approximate maximum independent set algorithm to identify the first elimination group [LiSaad].

### `sparse_linear_algebra_library_type = SUITE_SPARSE`

linear_solver_ordering_type = AMD

A constrained Approximate Minimum Degree (CAMD) ordering is used where the parameter blocks in the lowest numbered group are eliminated first, and then the parameter blocks in the next lowest numbered group and so on. Within each group, CAMD is free to order the parameter blocks as it chooses.

linear_solver_ordering_type = NESDIS

1. `linear_solver_type = SPARSE_NORMAL_CHOLESKY` or `linear_solver_type = CGNR` and `preconditioner_type = SUBSET`

   The value of `linear_solver_ordering` is ignored and a Nested Dissection algorithm is used to compute a fill reducing ordering.

2. `linear_solver_type = SPARSE_SCHUR/DENSE_SCHUR/ITERATIVE_SCHUR`

   ONLY the lowest group are used to compute the Schur complement, and Nested Dissection is used to compute a fill reducing ordering for the Schur Complement (or its preconditioner).

### `sparse_linear_algebra_library_type = EIGEN_SPARSE/ACCELERATE_SPARSE`

1. `linear_solver_type = SPARSE_NORMAL_CHOLESKY` or `linear_solver_type = CGNR` and `preconditioner_type = SUBSET`

   The value of `linear_solver_ordering` is ignored and `AMD` or `NESDIS` is used to compute a fill reducing ordering as requested by the user.

2. `linear_solver_type = SPARSE_SCHUR/DENSE_SCHUR/ITERATIVE_SCHUR`

   ONLY the lowest group are used to compute the Schur complement, and `AMD` or `NESID` is used to compute a fill reducing ordering for the Schur Complement (or its preconditioner) as requested by the user.

## `Solver::Options`

class Solver::Options

  `Solver::Options` controls the overall behavior of the solver. We list the various settings and their default values below.

bool Solver::Options::IsValid(std::string *error) const

  Validate the values in the options struct and returns true on success. If there is a problem, the method returns false with `error` containing a textual description of the cause.

MinimizerType Solver::Options::minimizer_type

  Default: `TRUST_REGION`

  Choose between `LINE_SEARCH` and `TRUST_REGION` algorithms. See Trust Region Methods and Line Search Methods for more details.

LineSearchDirectionType Solver::Options::line_search_direction_type

  Default: `LBFGS`

  Choices are `STEEPEST_DESCENT`, `NONLINEAR_CONJUGATE_GRADIENT`, `BFGS` and `LBFGS`.

  See Line Search Methods for more details.

LineSearchType Solver::Options::line_search_type

  Default: `WOLFE`

  Choices are `ARMIJO` and `WOLFE` (strong Wolfe conditions). Note that in order for the assumptions underlying the `BFGS` and `LBFGS` line search direction algorithms to be satisfied, the `WOLFE` line search must be used.

  See Line Search Methods for more details.

NonlinearConjugateGradientType Solver::Options::nonlinear_conjugate_gradient_type

  Default: `FLETCHER_REEVES`

  Choices are `FLETCHER_REEVES`, `POLAK_RIBIERE` and `HESTENES_STIEFEL`.

int Solver::Options::max_lbfgs_rank

  Default: `20`

  The LBFGS hessian approximation is a low rank approximation to the inverse of the Hessian matrix. The rank of the approximation determines (linearly) the space and time complexity of using the approximation. Higher the rank, the better is the quality of the approximation. The increase in quality is however is bounded for a number of reasons.

  1. The method only uses secant information and not actual derivatives. 2. The Hessian approximation is constrained to be positive definite.

  So increasing this rank to a large number will cost time and space complexity without the corresponding increase in solution quality. There are no hard and fast rules for choosing the maximum rank. The best choice usually requires some problem specific experimentation.

  For more theoretical and implementation details of the LBFGS method, please see [Nocedal].

bool Solver::Options::use_approximate_eigenvalue_bfgs_scaling

  > Default: `false`

  > As part of the `BFGS` update step / `LBFGS` right-multiply step, the initial inverse Hessian approximation is taken to be the Identity. However, [Oren] showed that using instead \(I * \gamma\), where \(\gamma\) is a scalar chosen to approximate an eigenvalue of the true inverse Hessian can result in improved convergence in a wide variety of cases. Setting `use_approximate_eigenvalue_bfgs_scaling` to true enables this scaling in `BFGS` (before first iteration) and `LBFGS` (at each iteration).

  > Precisely, approximate eigenvalue scaling equates to

  > \[\gamma = \frac{y_k' s_k}{y_k' y_k}\]

  > With:

  \[y_k = \nabla f_{k+1} - \nabla f_k\]
  \[s_k = x_{k+1} - x_k\]

  Where \(f()\) is the line search objective and \(x\) the vector of parameter values [NocedalWright].

  It is important to note that approximate eigenvalue scaling does not always improve convergence, and that it can in fact significantly degrade performance for certain classes of problem, which is why it is disabled by default. In particular it can degrade performance when the sensitivity of the problem to different parameters varies significantly, as in this case a single scalar factor fails to capture this variation and detrimentally downscales parts of the Jacobian approximation which correspond to low-sensitivity parameters. It can also reduce the robustness of the solution to errors in the Jacobians.

LineSearchIterpolationType Solver::Options::line_search_interpolation_type

  Default: `CUBIC`

  Degree of the polynomial used to approximate the objective function. Valid values are `BISECTION`, `QUADRATIC` and `CUBIC`.

double Solver::Options::min_line_search_step_size

  Default: `1e-9`

  The line search terminates if:

  \[\|\Delta x_k\|_\infty < \text{min_line_search_step_size}\]

  where \(\|\cdot\|_\infty\) refers to the max norm, and \(\Delta x_k\) is the step change in the parameter values at the \(k\)-th iteration.

double Solver::Options::line_search_sufficient_function_decrease

  Default: `1e-4`

  Solving the line search problem exactly is computationally prohibitive. Fortunately, line search based optimization algorithms can still guarantee convergence if instead of an exact solution, the line search algorithm returns a solution which decreases the value of the objective function sufficiently. More precisely, we are looking for a step size s.t.

  \[f(\text{step_size}) \le f(0) + \text{sufficient_decrease} * [f'(0) * \text{step_size}]\]

  This condition is known as the Armijo condition.

double Solver::Options::max_line_search_step_contraction

  Default: `1e-3`

  In each iteration of the line search,

  \[\text{new_step_size} >= \text{max_line_search_step_contraction} * \text{step_size}\]

  Note that by definition, for contraction:

  \[0 < \text{max_step_contraction} < \text{min_step_contraction} < 1\]

double Solver::Options::min_line_search_step_contraction

  Default: `0.6`

  In each iteration of the line search,

  \[\text{new_step_size} <= \text{min_line_search_step_contraction} * \text{step_size}\]

  Note that by definition, for contraction:

  \[0 < \text{max_step_contraction} < \text{min_step_contraction} < 1\]

int Solver::Options::max_num_line_search_step_size_iterations

  Default: `20`

  Maximum number of trial step size iterations during each line search, if a step size satisfying the search conditions cannot be found within this number of trials, the line search will stop.

  The minimum allowed value is 0 for trust region minimizer and 1 otherwise. If 0 is specified for the trust region minimizer, then line search will not be used when solving constrained optimization problems.

  As this is an ‘artificial’ constraint (one imposed by the user, not the underlying math), if `WOLFE` line search is being used, and points satisfying the Armijo sufficient (function) decrease condition have been found during the current search (in \(<=\) `max_num_line_search_step_size_iterations`). Then, the step size with the lowest function value which satisfies the Armijo condition will be returned as the new valid step, even though it does not satisfy the strong Wolfe conditions. This behaviour protects against early termination of the optimizer at a sub-optimal point.

int Solver::Options::max_num_line_search_direction_restarts

  Default: `5`

  Maximum number of restarts of the line search direction algorithm before terminating the optimization. Restarts of the line search direction algorithm occur when the current algorithm fails to produce a new descent direction. This typically indicates a numerical failure, or a breakdown in the validity of the approximations used.

double Solver::Options::line_search_sufficient_curvature_decrease

  Default: `0.9`

  The strong Wolfe conditions consist of the Armijo sufficient decrease condition, and an additional requirement that the step size be chosen s.t. the magnitude (‘strong’ Wolfe conditions) of the gradient along the search direction decreases sufficiently. Precisely, this second condition is that we seek a step size s.t.

  \[\|f'(\text{step_size})\| <= \text{sufficient_curvature_decrease} * \|f'(0)\|\]

  Where \(f()\) is the line search objective and \(f'()\) is the derivative of \(f\) with respect to the step size: \(\frac{d f}{d~\text{step size}}\).

double Solver::Options::max_line_search_step_expansion

  Default: `10.0`

  During the bracketing phase of a Wolfe line search, the step size is increased until either a point satisfying the Wolfe conditions is found, or an upper bound for a bracket containing a point satisfying the conditions is found. Precisely, at each iteration of the expansion:

  \[\text{new_step_size} <= \text{max_step_expansion} * \text{step_size}\]

  By definition for expansion

  \[\text{max_step_expansion} > 1.0\]

TrustRegionStrategyType Solver::Options::trust_region_strategy_type

  Default: `LEVENBERG_MARQUARDT`

  The trust region step computation algorithm used by Ceres. Currently `LEVENBERG_MARQUARDT` and `DOGLEG` are the two valid choices. See Levenberg-Marquardt and Dogleg for more details.

DoglegType Solver::Options::dogleg_type

  Default: `TRADITIONAL_DOGLEG`

  Ceres supports two different dogleg strategies. `TRADITIONAL_DOGLEG` method by Powell and the `SUBSPACE_DOGLEG` method described by [ByrdSchnabel] . See Dogleg for more details.

bool Solver::Options::use_nonmonotonic_steps

  Default: `false`

  Relax the requirement that the trust-region algorithm take strictly decreasing steps. See Non-monotonic Steps for more details.

int Solver::Options::max_consecutive_nonmonotonic_steps

  Default: `5`

  The window size used by the step selection algorithm to accept non-monotonic steps.

int Solver::Options::max_num_iterations

  Default: `50`

  Maximum number of iterations for which the solver should run.

double Solver::Options::max_solver_time_in_seconds

  Default: `1e9`

  Maximum amount of time for which the solver should run.

int Solver::Options::num_threads

  Default: `1`

  Number of threads used by Ceres to evaluate the Jacobian.

double Solver::Options::initial_trust_region_radius

  Default: `1e4`

  The size of the initial trust region. When the `LEVENBERG_MARQUARDT` strategy is used, the reciprocal of this number is the initial regularization parameter.

double Solver::Options::max_trust_region_radius

  Default: `1e16`

  The trust region radius is not allowed to grow beyond this value.

double Solver::Options::min_trust_region_radius

  Default: `1e-32`

  The solver terminates, when the trust region becomes smaller than this value.

double Solver::Options::min_relative_decrease

  Default: `1e-3`

  Lower threshold for relative decrease before a trust-region step is accepted.

double Solver::Options::min_lm_diagonal

  Default: `1e-6`

  The `LEVENBERG_MARQUARDT` strategy, uses a diagonal matrix to regularize the trust region step. This is the lower bound on the values of this diagonal matrix.

double Solver::Options::max_lm_diagonal

  Default: `1e32`

  The `LEVENBERG_MARQUARDT` strategy, uses a diagonal matrix to regularize the trust region step. This is the upper bound on the values of this diagonal matrix.

int Solver::Options::max_num_consecutive_invalid_steps

  Default: `5`

  The step returned by a trust region strategy can sometimes be numerically invalid, usually because of conditioning issues. Instead of crashing or stopping the optimization, the optimizer can go ahead and try solving with a smaller trust region/better conditioned problem. This parameter sets the number of consecutive retries before the minimizer gives up.

double Solver::Options::function_tolerance

  Default: `1e-6`

  Solver terminates if

  \[\frac{|\Delta \text{cost}|}{\text{cost}} <= \text{function_tolerance}\]

  where, \(\Delta \text{cost}\) is the change in objective function value (up or down) in the current iteration of Levenberg-Marquardt.

double Solver::Options::gradient_tolerance

  Default: `1e-10`

  Solver terminates if

  \[\|x - \Pi \boxplus(x, -g(x))\|_\infty <= \text{gradient_tolerance}\]

  where \(\|\cdot\|_\infty\) refers to the max norm, \(\Pi\) is projection onto the bounds constraints and \(\boxplus\) is Plus operation for the overall manifold associated with the parameter vector.

double Solver::Options::parameter_tolerance

  Default: `1e-8`

  Solver terminates if

  \[\|\Delta x\| <= (\|x\| + \text{parameter_tolerance}) * \text{parameter_tolerance}\]

  where \(\Delta x\) is the step computed by the linear solver in the current iteration.

LinearSolverType Solver::Options::linear_solver_type

  Default: `SPARSE_NORMAL_CHOLESKY` / `DENSE_QR`

  Type of linear solver used to compute the solution to the linear least squares problem in each iteration of the Levenberg-Marquardt algorithm. If Ceres is built with support for `SuiteSparse` or `Accelerate` or `Eigen`’s sparse Cholesky factorization, the default is `SPARSE_NORMAL_CHOLESKY`, it is `DENSE_QR` otherwise.

PreconditionerType Solver::Options::preconditioner_type

  Default: `JACOBI`

  The preconditioner used by the iterative linear solver. The default is the block Jacobi preconditioner. Valid values are (in increasing order of complexity) `IDENTITY`, `JACOBI`, `SCHUR_JACOBI`, `CLUSTER_JACOBI`, `CLUSTER_TRIDIAGONAL`, `SUBSET` and `SCHUR_POWER_SERIES_EXPANSION`. See Preconditioners for more details.

VisibilityClusteringType Solver::Options::visibility_clustering_type

  Default: `CANONICAL_VIEWS`

  Type of clustering algorithm to use when constructing a visibility based preconditioner. The original visibility based preconditioning paper and implementation only used the canonical views algorithm.

  This algorithm gives high quality results but for large dense graphs can be particularly expensive. As its worst case complexity is cubic in size of the graph.

  Another option is to use `SINGLE_LINKAGE` which is a simple thresholded single linkage clustering algorithm that only pays attention to tightly coupled blocks in the Schur complement. This is a fast algorithm that works well.

  The optimal choice of the clustering algorithm depends on the sparsity structure of the problem, but generally speaking we recommend that you try `CANONICAL_VIEWS` first and if it is too expensive try `SINGLE_LINKAGE`.

std::unordered_set<ResidualBlockId> Solver::Options::residual_blocks_for_subset_preconditioner

  `SUBSET` preconditioner is a preconditioner for problems with general sparsity. Given a subset of residual blocks of a problem, it uses the corresponding subset of the rows of the Jacobian to construct a preconditioner.

  Suppose the Jacobian \(J\) has been horizontally partitioned as

  > \[\begin{split}J = \begin{bmatrix} P \\ Q \end{bmatrix}\end{split}\]

  Where, \(Q\) is the set of rows corresponding to the residual blocks in `Solver::Options::residual_blocks_for_subset_preconditioner`. The preconditioner is the matrix \((Q^\top Q)^{-1}\).

  The efficacy of the preconditioner depends on how well the matrix \(Q\) approximates \(J^\top J\), or how well the chosen residual blocks approximate the full problem.

  If `Solver::Options::preconditioner_type == SUBSET`, then `residual_blocks_for_subset_preconditioner` must be non-empty.

DenseLinearAlgebraLibrary Solver::Options::dense_linear_algebra_library_type

  Default: `EIGEN`

  Ceres supports using multiple dense linear algebra libraries for dense matrix factorizations. Currently `EIGEN`, `LAPACK` and `CUDA` are the valid choices. `EIGEN` is always available, `LAPACK` refers to the system `BLAS + LAPACK` library which may or may not be available. `CUDA` refers to Nvidia’s GPU based dense linear algebra library which may or may not be available.

  This setting affects the `DENSE_QR`, `DENSE_NORMAL_CHOLESKY` and `DENSE_SCHUR` solvers. For small to moderate sized problem `EIGEN` is a fine choice but for large problems, an optimized `LAPACK + BLAS` or `CUDA` implementation can make a substantial difference in performance.

SparseLinearAlgebraLibrary Solver::Options::sparse_linear_algebra_library_type

  Default: The highest available according to: `SUITE_SPARSE` > `ACCELERATE_SPARSE` > `EIGEN_SPARSE` > `NO_SPARSE`

  Ceres supports the use of three sparse linear algebra libraries, `SuiteSparse`, which is enabled by setting this parameter to `SUITE_SPARSE`, `Acclerate`, which can be selected by setting this parameter to `ACCELERATE_SPARSE` and `Eigen` which is enabled by setting this parameter to `EIGEN_SPARSE`. Lastly, `NO_SPARSE` means that no sparse linear solver should be used; note that this is irrespective of whether Ceres was compiled with support for one.

  `SuiteSparse` is a sophisticated sparse linear algebra library and should be used in general. On MacOS you may want to use the `Accelerate` framework.

  If your needs/platforms prevent you from using `SuiteSparse`, consider using the sparse linear algebra routines in `Eigen`. The sparse Cholesky algorithms currently included with `Eigen` are not as sophisticated as the ones in `SuiteSparse` and `Accelerate` and as a result its performance is considerably worse.

LinearSolverOrderingType Solver::Options::linear_solver_ordering_type

  Default: `AMD`

  > The order in which variables are eliminated in a linear solver can have a significant impact on the efficiency and accuracy of the method. e.g., when doing sparse Cholesky factorization, there are matrices for which a good ordering will give a Cholesky factor with \(O(n)\) storage, where as a bad ordering will result in an completely dense factor.

  > Sparse direct solvers like `SPARSE_NORMAL_CHOLESKY` and `SPARSE_SCHUR` use a fill reducing ordering of the columns and rows of the matrix being factorized before computing the numeric factorization.

  > This enum controls the type of algorithm used to compute this fill reducing ordering. There is no single algorithm that works on all matrices, so determining which algorithm works better is a matter of empirical experimentation.

std::shared_ptr<ParameterBlockOrdering> Solver::Options::linear_solver_ordering

  Default: `nullptr`

  An instance of the ordering object informs the solver about the desired order in which parameter blocks should be eliminated by the linear solvers.

  If `nullptr`, the solver is free to choose an ordering that it thinks is best.

  See Ordering for more details.

bool Solver::Options::use_explicit_schur_complement

  Default: `false`

  Use an explicitly computed Schur complement matrix with `ITERATIVE_SCHUR`.

  By default this option is disabled and `ITERATIVE_SCHUR` evaluates evaluates matrix-vector products between the Schur complement and a vector implicitly by exploiting the algebraic expression for the Schur complement.

  The cost of this evaluation scales with the number of non-zeros in the Jacobian.

  For small to medium sized problems there is a sweet spot where computing the Schur complement is cheap enough that it is much more efficient to explicitly compute it and use it for evaluating the matrix-vector products.

  Note

  This option can only be used with the `SCHUR_JACOBI` preconditioner.

bool Solver::Options::dynamic_sparsity

  Default: `false`

  Some non-linear least squares problems are symbolically dense but numerically sparse. i.e. at any given state only a small number of Jacobian entries are non-zero, but the position and number of non-zeros is different depending on the state. For these problems it can be useful to factorize the sparse jacobian at each solver iteration instead of including all of the zero entries in a single general factorization.

  If your problem does not have this property (or you do not know), then it is probably best to keep this false, otherwise it will likely lead to worse performance.

  This setting only affects the SPARSE_NORMAL_CHOLESKY solver.

bool Solver::Options::use_mixed_precision_solves

  Default: `false`

  If true, the Gauss-Newton matrix is computed in double precision, but its factorization is computed in single precision. This can result in significant time and memory savings at the cost of some accuracy in the Gauss-Newton step. Iterative refinement is used to recover some of this accuracy back.

  If `use_mixed_precision_solves` is true, we recommend setting `max_num_refinement_iterations` to 2-3.

  See Mixed Precision Solves for more details.

int Solver::Options::max_num_refinement_iterations

  Default: `0`

  Number steps of the iterative refinement process to run when computing the Gauss-Newton step, see `Solver::Options::use_mixed_precision_solves`.

int Solver::Options::min_linear_solver_iterations

  Default: `0`

  Minimum number of iterations used by the linear solver. This only makes sense when the linear solver is an iterative solver, e.g., `ITERATIVE_SCHUR` or `CGNR`.

int Solver::Options::max_linear_solver_iterations

  Default: `500`

  Minimum number of iterations used by the linear solver. This only makes sense when the linear solver is an iterative solver, e.g., `ITERATIVE_SCHUR` or `CGNR`.

int Solver::Options::max_num_spse_iterations

  Default: 5

  Maximum number of iterations performed by `SCHUR_POWER_SERIES_EXPANSION`. Each iteration corresponds to one more term in the power series expansion od the inverse of the Schur complement. This value controls the maximum number of iterations whether it is used as a preconditioner or just to initialize the solution for `ITERATIVE_SCHUR`.

bool Solver : Options::use_spse_initialization

  Default: `false`

  Use Schur power series expansion to initialize the solution for `ITERATIVE_SCHUR`. This option can be set `true` regardless of what preconditioner is being used.

double Solver::Options::spse_tolerance

  Default: 0.1

  When `use_spse_initialization` is `true`, this parameter along with `max_num_spse_iterations` controls the number of `SCHUR_POWER_SERIES_EXPANSION` iterations performed for initialization. It is not used to control the preconditioner.

double Solver::Options::eta

  Default: `1e-1`

  Forcing sequence parameter. The truncated Newton solver uses this number to control the relative accuracy with which the Newton step is computed. This constant is passed to `ConjugateGradientsSolver` which uses it to terminate the iterations when

  \[\frac{Q_i - Q_{i-1}}{Q_i} < \frac{\eta}{i}\]

bool Solver::Options::jacobi_scaling

  Default: `true`

  `true` means that the Jacobian is scaled by the norm of its columns before being passed to the linear solver. This improves the numerical conditioning of the normal equations.

bool Solver::Options::use_inner_iterations

  Default: `false`

  Use a non-linear version of a simplified variable projection algorithm. Essentially this amounts to doing a further optimization on each Newton/Trust region step using a coordinate descent algorithm. For more details, see Inner Iterations.

  Note Inner iterations cannot be used with `Problem` objects that have an `EvaluationCallback` associated with them.

std::shared_ptr<ParameterBlockOrdering> Solver::Options::inner_iteration_ordering

  Default: `nullptr`

  If `Solver::Options::use_inner_iterations` true, then the user has two choices.

  1. Let the solver heuristically decide which parameter blocks to optimize in each inner iteration. To do this, set `Solver::Options::inner_iteration_ordering` to `nullptr`.

  2. Specify a collection of of ordered independent sets. The lower numbered groups are optimized before the higher number groups during the inner optimization phase. Each group must be an independent set. Not all parameter blocks need to be included in the ordering.

  See Ordering for more details.

double Solver::Options::inner_iteration_tolerance

  Default: `1e-3`

  Generally speaking, inner iterations make significant progress in the early stages of the solve and then their contribution drops down sharply, at which point the time spent doing inner iterations is not worth it.

  Once the relative decrease in the objective function due to inner iterations drops below `inner_iteration_tolerance`, the use of inner iterations in subsequent trust region minimizer iterations is disabled.

LoggingType Solver::Options::logging_type

  Default: `PER_MINIMIZER_ITERATION`

  Valid values are `SILENT` and `PER_MINIMIZER_ITERATION`.

bool Solver::Options::minimizer_progress_to_stdout

  Default: `false`

  By default the Minimizer’s progress is logged to `STDERR` depending on the `vlog` level. If this flag is set to true, and `Solver::Options::logging_type` is not `SILENT`, the logging output is sent to `STDOUT`.

  For `TRUST_REGION_MINIMIZER` the progress display looks like

      iter      cost      cost_change  |gradient|   |step|    tr_ratio  tr_radius  ls_iter  iter_time  total_time
         0  4.185660e+06    0.00e+00    1.09e+08   0.00e+00   0.00e+00  1.00e+04       0    7.59e-02    3.37e-01
         1  1.062590e+05    4.08e+06    8.99e+06   5.36e+02   9.82e-01  3.00e+04       1    1.65e-01    5.03e-01
         2  4.992817e+04    5.63e+04    8.32e+06   3.19e+02   6.52e-01  3.09e+04       1    1.45e-01    6.48e-01

  Here

  1. `cost` is the value of the objective function.

  2. `cost_change` is the change in the value of the objective function if the step computed in this iteration is accepted.

  3. `|gradient|` is the max norm of the gradient.

  4. `|step|` is the change in the parameter vector.

  5. `tr_ratio` is the ratio of the actual change in the objective function value to the change in the value of the trust region model.

  6. `tr_radius` is the size of the trust region radius.

  7. `ls_iter` is the number of linear solver iterations used to compute the trust region step. For direct/factorization based solvers it is always 1, for iterative solvers like `ITERATIVE_SCHUR` it is the number of iterations of the Conjugate Gradients algorithm.

  8. `iter_time` is the time take by the current iteration.

  9. `total_time` is the total time taken by the minimizer.

  For `LINE_SEARCH_MINIMIZER` the progress display looks like

      0: f: 2.317806e+05 d: 0.00e+00 g: 3.19e-01 h: 0.00e+00 s: 0.00e+00 e:  0 it: 2.98e-02 tt: 8.50e-02
      1: f: 2.312019e+05 d: 5.79e+02 g: 3.18e-01 h: 2.41e+01 s: 1.00e+00 e:  1 it: 4.54e-02 tt: 1.31e-01
      2: f: 2.300462e+05 d: 1.16e+03 g: 3.17e-01 h: 4.90e+01 s: 2.54e-03 e:  1 it: 4.96e-02 tt: 1.81e-01

  Here

  1. `f` is the value of the objective function.

  2. `d` is the change in the value of the objective function if the step computed in this iteration is accepted.

  3. `g` is the max norm of the gradient.

  4. `h` is the change in the parameter vector.

  5. `s` is the optimal step length computed by the line search.

  6. `it` is the time take by the current iteration.

  7. `tt` is the total time taken by the minimizer.

std::vector<int> Solver::Options::trust_region_minimizer_iterations_to_dump

  Default: `empty`

  List of iterations at which the trust region minimizer should dump the trust region problem. Useful for testing and benchmarking. If `empty`, no problems are dumped.

std::string Solver::Options::trust_region_problem_dump_directory

  Default: `/tmp`

  > Directory to which the problems should be written to. Should be non-empty if `Solver::Options::trust_region_minimizer_iterations_to_dump` is non-empty and `Solver::Options::trust_region_problem_dump_format_type` is not `CONSOLE`.

DumpFormatType Solver::Options::trust_region_problem_dump_format_type

  Default: `TEXTFILE`

  The format in which trust region problems should be logged when `Solver::Options::trust_region_minimizer_iterations_to_dump` is non-empty. There are three options:

  - `CONSOLE` prints the linear least squares problem in a human

      readable format to `stderr`. The Jacobian is printed as a dense matrix. The vectors \(D\), \(x\) and \(f\) are printed as dense vectors. This should only be used for small problems.

  - `TEXTFILE` Write out the linear least squares problem to the directory pointed to by `Solver::Options::trust_region_problem_dump_directory` as text files which can be read into `MATLAB/Octave`. The Jacobian is dumped as a text file containing \((i,j,s)\) triplets, the vectors \(D\), x and f are dumped as text files containing a list of their values.

    A `MATLAB/Octave` script called `ceres_solver_iteration_???.m` is also output, which can be used to parse and load the problem into memory.

bool Solver::Options::check_gradients

  Default: `false`

  Check all Jacobians computed by each residual block with finite differences. This is expensive since it involves computing the derivative by normal means (e.g. user specified, autodiff, etc), then also computing it using finite differences. The results are compared, and if they differ substantially, the optimization fails and the details are stored in the solver summary.

double Solver::Options::gradient_check_relative_precision

  Default: `1e-8`

  Precision to check for in the gradient checker. If the relative difference between an element in a Jacobian exceeds this number, then the Jacobian for that cost term is dumped.

double Solver::Options::gradient_check_numeric_derivative_relative_step_size

  Default: `1e-6`

  Note

  This option only applies to the numeric differentiation used for checking the user provided derivatives when when Solver::Options::check_gradients is true. If you are using `NumericDiffCostFunction` and are interested in changing the step size for numeric differentiation in your cost function, please have a look at `NumericDiffOptions`.

  Relative shift used for taking numeric derivatives when Solver::Options::check_gradients is true.

  For finite differencing, each dimension is evaluated at slightly shifted values, e.g., for forward differences, the numerical derivative is

  \[\begin{split}\delta &= gradient\_check\_numeric\_derivative\_relative\_step\_size\\ \Delta f &= \frac{f((1 + \delta) x) - f(x)}{\delta x}\end{split}\]

  The finite differencing is done along each dimension. The reason to use a relative (rather than absolute) step size is that this way, numeric differentiation works for functions where the arguments are typically large (e.g. \(10^9\)) and when the values are small (e.g. \(10^{-5}\)). It is possible to construct torture cases which break this finite difference heuristic, but they do not come up often in practice.

bool Solver::Options::update_state_every_iteration

  Default: `false`

  If `update_state_every_iteration` is `true`, then Ceres Solver will guarantee that at the end of every iteration and before any user `IterationCallback` is called, the parameter blocks are updated to the current best solution found by the solver. Thus the IterationCallback can inspect the values of the parameter blocks for purposes of computation, visualization or termination.

  If `update_state_every_iteration` is `false` then there is no such guarantee, and user provided `IterationCallback` s should not expect to look at the parameter blocks and interpret their values.

std::vector<IterationCallback*> Solver::Options::callbacks

  Default: `empty`

  Callbacks that are executed at the end of each iteration of the minimizer. They are executed in the order that they are specified in this vector.

  By default, parameter blocks are updated only at the end of the optimization, i.e., when the minimizer terminates. This means that by default, if an `IterationCallback` inspects the parameter blocks, they will not see them changing in the course of the optimization.

  To tell Ceres to update the parameter blocks at the end of each iteration and before calling the user’s callback, set `Solver::Options::update_state_every_iteration` to `true`.

  See examples/iteration_callback_example.cc for an example of an `IterationCallback` that uses `Solver::Options::update_state_every_iteration` to log changes to the parameter blocks over the course of the optimization.

  The solver does NOT take ownership of these pointers.

## `ParameterBlockOrdering`

class ParameterBlockOrdering

  `ParameterBlockOrdering` is a class for storing and manipulating an ordered collection of groups/sets with the following semantics:

  Group IDs are non-negative integer values. Elements are any type that can serve as a key in a map or an element of a set.

  An element can only belong to one group at a time. A group may contain an arbitrary number of elements.

  Groups are ordered by their group id.

bool ParameterBlockOrdering::AddElementToGroup(const double *element, const int group)

  Add an element to a group. If a group with this id does not exist, one is created. This method can be called any number of times for the same element. Group ids should be non-negative numbers. Return value indicates if adding the element was a success.

void ParameterBlockOrdering::Clear()

  Clear the ordering.

bool ParameterBlockOrdering::Remove(const double *element)

  Remove the element, no matter what group it is in. If the element is not a member of any group, calling this method will result in a crash. Return value indicates if the element was actually removed.

void ParameterBlockOrdering::Reverse()

  Reverse the order of the groups in place.

int ParameterBlockOrdering::GroupId(const double *element) const

  Return the group id for the element. If the element is not a member of any group, return -1.

bool ParameterBlockOrdering::IsMember(const double *element) const

  True if there is a group containing the parameter block.

int ParameterBlockOrdering::GroupSize(const int group) const

  This function always succeeds, i.e., implicitly there exists a group for every integer.

int ParameterBlockOrdering::NumElements() const

  Number of elements in the ordering.

int ParameterBlockOrdering::NumGroups() const

  Number of groups with one or more elements.

## `IterationSummary`

class IterationSummary

  `IterationSummary` describes the state of the minimizer at the end of each iteration.

int IterationSummary::iteration

  Current iteration number.

bool IterationSummary::step_is_valid

  Step was numerically valid, i.e., all values are finite and the step reduces the value of the linearized model.

  > Note: `IterationSummary::step_is_valid` is false when `IterationSummary::iteration` = 0.

bool IterationSummary::step_is_nonmonotonic

  Step did not reduce the value of the objective function sufficiently, but it was accepted because of the relaxed acceptance criterion used by the non-monotonic trust region algorithm.

  Note: `IterationSummary::step_is_nonmonotonic` is false when when `IterationSummary::iteration` = 0.

bool IterationSummary::step_is_successful

  Whether or not the minimizer accepted this step or not.

  If the ordinary trust region algorithm is used, this means that the relative reduction in the objective function value was greater than `Solver::Options::min_relative_decrease`. However, if the non-monotonic trust region algorithm is used (`Solver::Options::use_nonmonotonic_steps` = true), then even if the relative decrease is not sufficient, the algorithm may accept the step and the step is declared successful.

  Note: `IterationSummary::step_is_successful` is false when when `IterationSummary::iteration` = 0.

double IterationSummary::cost

  Value of the objective function.

double IterationSummary::cost_change

  Change in the value of the objective function in this iteration. This can be positive or negative.

double IterationSummary::gradient_max_norm

  Infinity norm of the gradient vector.

double IterationSummary::gradient_norm

  2-norm of the gradient vector.

double IterationSummary::step_norm

  2-norm of the size of the step computed in this iteration.

double IterationSummary::relative_decrease

  For trust region algorithms, the ratio of the actual change in cost and the change in the cost of the linearized approximation.

  This field is not used when a linear search minimizer is used.

double IterationSummary::trust_region_radius

  Size of the trust region at the end of the current iteration. For the Levenberg-Marquardt algorithm, the regularization parameter is 1.0 / `IterationSummary::trust_region_radius`.

double IterationSummary::eta

  For the inexact step Levenberg-Marquardt algorithm, this is the relative accuracy with which the step is solved. This number is only applicable to the iterative solvers capable of solving linear systems inexactly. Factorization-based exact solvers always have an eta of 0.0.

double IterationSummary::step_size

  Step sized computed by the line search algorithm.

  This field is not used when a trust region minimizer is used.

int IterationSummary::line_search_function_evaluations

  Number of function evaluations used by the line search algorithm.

  This field is not used when a trust region minimizer is used.

int IterationSummary::linear_solver_iterations

  Number of iterations taken by the linear solver to solve for the trust region step.

  Currently this field is not used when a line search minimizer is used.

double IterationSummary::iteration_time_in_seconds

  Time (in seconds) spent inside the minimizer loop in the current iteration.

double IterationSummary::step_solver_time_in_seconds

  Time (in seconds) spent inside the trust region step solver.

double IterationSummary::cumulative_time_in_seconds

  Time (in seconds) since the user called Solve().

## `IterationCallback`

class IterationCallback

  > Interface for specifying callbacks that are executed at the end of each iteration of the minimizer.

  >     class IterationCallback {
  >      public:
  >       virtual ~IterationCallback() {}
  >       virtual CallbackReturnType operator()(const IterationSummary& summary) = 0;
  >     };

  The solver uses the return value of `operator()` to decide whether to continue solving or to terminate. The user can return three values.

  1. `SOLVER_ABORT` indicates that the callback detected an abnormal situation. The solver returns without updating the parameter blocks (unless `Solver::Options::update_state_every_iteration` is set true). Solver returns with `Solver::Summary::termination_type` set to `USER_FAILURE`.

  2. `SOLVER_TERMINATE_SUCCESSFULLY` indicates that there is no need to optimize anymore (some user specified termination criterion has been met). Solver returns with `Solver::Summary::termination_type`` set to `USER_SUCCESS`.

  3. `SOLVER_CONTINUE` indicates that the solver should continue optimizing.

  The return values can be used to implement custom termination criterion that supercede the iteration/time/tolerance based termination implemented by Ceres.

  For example, the following `IterationCallback` is used internally by Ceres to log the progress of the optimization.

      class LoggingCallback : public IterationCallback {
       public:
        explicit LoggingCallback(bool log_to_stdout)
            : log_to_stdout_(log_to_stdout) {}

        ~LoggingCallback() {}

        CallbackReturnType operator()(const IterationSummary& summary) {
          const char* kReportRowFormat =
              "% 4d: f:% 8e d:% 3.2e g:% 3.2e h:% 3.2e "
              "rho:% 3.2e mu:% 3.2e eta:% 3.2e li:% 3d";
          string output = StringPrintf(kReportRowFormat,
                                       summary.iteration,
                                       summary.cost,
                                       summary.cost_change,
                                       summary.gradient_max_norm,
                                       summary.step_norm,
                                       summary.relative_decrease,
                                       summary.trust_region_radius,
                                       summary.eta,
                                       summary.linear_solver_iterations);
          if (log_to_stdout_) {
            cout << output << endl;
          } else {
            VLOG(1) << output;
          }
          return SOLVER_CONTINUE;
        }

       private:
        const bool log_to_stdout_;
      };

  See examples/evaluation_callback_example.cc for another example that uses `Solver::Options::update_state_every_iteration` to log changes to the parameter blocks over the course of the optimization.

## `CRSMatrix`

class CRSMatrix

  A compressed row sparse matrix used primarily for communicating the Jacobian matrix to the user.

int CRSMatrix::num_rows

  Number of rows.

int CRSMatrix::num_cols

  Number of columns.

std::vector<int> CRSMatrix::rows

  `CRSMatrix::rows` is a `CRSMatrix::num_rows` + 1 sized array that points into the `CRSMatrix::cols` and `CRSMatrix::values` array.

std::vector<int> CRSMatrix::cols

  `CRSMatrix::cols` contain as many entries as there are non-zeros in the matrix.

  For each row `i`, `cols[rows[i]]` … `cols[rows[i + 1] - 1]` are the indices of the non-zero columns of row `i`.

std::vector<double> CRSMatrix::values

  `CRSMatrix::values` contain as many entries as there are non-zeros in the matrix.

  For each row `i`, `values[rows[i]]` … `values[rows[i + 1] - 1]` are the values of the non-zero columns of row `i`.

e.g., consider the 3x4 sparse matrix

    0 10  0  4
    0  2 -3  2
    1  2  0  0

The three arrays will be:

             -row0-  ---row1---  -row2-
    rows   = [ 0,      2,          5,     7]
    cols   = [ 1,  3,  1,  2,  3,  0,  1]
    values = [10,  4,  2, -3,  2,  1,  2]

## `Solver::Summary`

class Solver::Summary

  Summary of the various stages of the solver after termination.

std::string Solver::Summary::BriefReport() const

  A brief one line description of the state of the solver after termination.

std::string Solver::Summary::FullReport() const

  A full multiline description of the state of the solver after termination.

bool Solver::Summary::IsSolutionUsable() const

  Whether the solution returned by the optimization algorithm can be relied on to be numerically sane. This will be the case if Solver::Summary:termination_type is set to CONVERGENCE, USER_SUCCESS or NO_CONVERGENCE, i.e., either the solver converged by meeting one of the convergence tolerances or because the user indicated that it had converged or it ran to the maximum number of iterations or time.

MinimizerType Solver::Summary::minimizer_type

  Type of minimization algorithm used.

TerminationType Solver::Summary::termination_type

  The cause of the minimizer terminating.

std::string Solver::Summary::message

  Reason why the solver terminated.

double Solver::Summary::initial_cost

  Cost of the problem (value of the objective function) before the optimization.

double Solver::Summary::final_cost

  Cost of the problem (value of the objective function) after the optimization.

double Solver::Summary::fixed_cost

  The part of the total cost that comes from residual blocks that were held fixed by the preprocessor because all the parameter blocks that they depend on were fixed.

std::vector<IterationSummary> Solver::Summary::iterations

  `IterationSummary` for each minimizer iteration in order.

int Solver::Summary::num_successful_steps

  Number of minimizer iterations in which the step was accepted. Unless `Solver::Options::use_nonmonotonic_steps` is true this is also the number of steps in which the objective function value/cost went down.

int Solver::Summary::num_unsuccessful_steps

  Number of minimizer iterations in which the step was rejected either because it did not reduce the cost enough or the step was not numerically valid.

int Solver::Summary::num_inner_iteration_steps

  Number of times inner iterations were performed.

int Solver::Summary::num_line_search_steps

  Total number of iterations inside the line search algorithm across all invocations. We call these iterations “steps” to distinguish them from the outer iterations of the line search and trust region minimizer algorithms which call the line search algorithm as a subroutine.

double Solver::Summary::preprocessor_time_in_seconds

  Time (in seconds) spent in the preprocessor.

double Solver::Summary::minimizer_time_in_seconds

  Time (in seconds) spent in the minimizer.

double Solver::Summary::postprocessor_time_in_seconds

  Time (in seconds) spent in the post processor.

double Solver::Summary::total_time_in_seconds

  Time (in seconds) spent in the solver.

double Solver::Summary::linear_solver_time_in_seconds

  Time (in seconds) spent in the linear solver computing the trust region step.

int Solver::Summary::num_linear_solves

  Number of times the Newton step was computed by solving a linear system. This does not include linear solves used by inner iterations.

double Solver::Summary::residual_evaluation_time_in_seconds

  Time (in seconds) spent evaluating the residual vector.

int Solver::Summary::num_residual_evaluations

  Number of times only the residuals were evaluated.

double Solver::Summary::jacobian_evaluation_time_in_seconds

  Time (in seconds) spent evaluating the Jacobian matrix.

int Solver::Summary::num_jacobian_evaluations

  Number of times only the Jacobian and the residuals were evaluated.

double Solver::Summary::inner_iteration_time_in_seconds

  Time (in seconds) spent doing inner iterations.

int Solver::Summary::num_parameter_blocks

  Number of parameter blocks in the problem.

int Solver::Summary::num_parameters

  Number of parameters in the problem.

int Solver::Summary::num_effective_parameters

  Dimension of the tangent space of the problem (or the number of columns in the Jacobian for the problem). This is different from `Solver::Summary::num_parameters` if a parameter block is associated with a `Manifold`.

int Solver::Summary::num_residual_blocks

  Number of residual blocks in the problem.

int Solver::Summary::num_residuals

  Number of residuals in the problem.

int Solver::Summary::num_parameter_blocks_reduced

  Number of parameter blocks in the problem after the inactive and constant parameter blocks have been removed. A parameter block is inactive if no residual block refers to it.

int Solver::Summary::num_parameters_reduced

  Number of parameters in the reduced problem.

int Solver::Summary::num_effective_parameters_reduced

  Dimension of the tangent space of the reduced problem (or the number of columns in the Jacobian for the reduced problem). This is different from `Solver::Summary::num_parameters_reduced` if a parameter block in the reduced problem is associated with a `Manifold`.

int Solver::Summary::num_residual_blocks_reduced

  Number of residual blocks in the reduced problem.

int Solver::Summary::num_residuals_reduced

  Number of residuals in the reduced problem.

int Solver::Summary::num_threads_given

  Number of threads specified by the user for Jacobian and residual evaluation.

int Solver::Summary::num_threads_used

  Number of threads actually used by the solver for Jacobian and residual evaluation.

LinearSolverType Solver::Summary::linear_solver_type_given

  Type of the linear solver requested by the user.

LinearSolverType Solver::Summary::linear_solver_type_used

  Type of the linear solver actually used. This may be different from `Solver::Summary::linear_solver_type_given` if Ceres determines that the problem structure is not compatible with the linear solver requested or if the linear solver requested by the user is not available, e.g. The user requested SPARSE_NORMAL_CHOLESKY but no sparse linear algebra library was available.

std::vector<int> Solver::Summary::linear_solver_ordering_given

  Size of the elimination groups given by the user as hints to the linear solver.

std::vector<int> Solver::Summary::linear_solver_ordering_used

  Size of the parameter groups used by the solver when ordering the columns of the Jacobian. This maybe different from `Solver::Summary::linear_solver_ordering_given` if the user left `Solver::Summary::linear_solver_ordering_given` blank and asked for an automatic ordering, or if the problem contains some constant or inactive parameter blocks.

std::string Solver::Summary::schur_structure_given

  For Schur type linear solvers, this string describes the template specialization which was detected in the problem and should be used.

std::string Solver::Summary::schur_structure_used

  For Schur type linear solvers, this string describes the template specialization that was actually instantiated and used. The reason this will be different from `Solver::Summary::schur_structure_given` is because the corresponding template specialization does not exist.

  Template specializations can be added to ceres by editing `internal/ceres/generate_template_specializations.py`

bool Solver::Summary::inner_iterations_given

  True if the user asked for inner iterations to be used as part of the optimization.

bool Solver::Summary::inner_iterations_used

  True if the user asked for inner iterations to be used as part of the optimization and the problem structure was such that they were actually performed. For example, in a problem with just one parameter block, inner iterations are not performed.

std::vector<int> Solver::Summary::inner_iteration_ordering_given

  Size of the parameter groups given by the user for performing inner iterations.

std::vector<int> Solver::Summary::inner_iteration_ordering_used

  Size of the parameter groups given used by the solver for performing inner iterations. This maybe different from `Solver::Summary::inner_iteration_ordering_given` if the user left `Solver::Summary::inner_iteration_ordering_given` blank and asked for an automatic ordering, or if the problem contains some constant or inactive parameter blocks.

PreconditionerType Solver::Summary::preconditioner_type_given

  Type of the preconditioner requested by the user.

PreconditionerType Solver::Summary::preconditioner_type_used

  Type of the preconditioner actually used. This may be different from `Solver::Summary::linear_solver_type_given` if Ceres determines that the problem structure is not compatible with the linear solver requested or if the linear solver requested by the user is not available.

VisibilityClusteringType Solver::Summary::visibility_clustering_type

  Type of clustering algorithm used for visibility based preconditioning. Only meaningful when the `Solver::Summary::preconditioner_type_used` is `CLUSTER_JACOBI` or `CLUSTER_TRIDIAGONAL`.

TrustRegionStrategyType Solver::Summary::trust_region_strategy_type

  Type of trust region strategy.

DoglegType Solver::Summary::dogleg_type

  Type of dogleg strategy used for solving the trust region problem.

DenseLinearAlgebraLibraryType Solver::Summary::dense_linear_algebra_library_type

  Type of the dense linear algebra library used.

SparseLinearAlgebraLibraryType Solver::Summary::sparse_linear_algebra_library_type

  Type of the sparse linear algebra library used.

LineSearchDirectionType Solver::Summary::line_search_direction_type

  Type of line search direction used.

LineSearchType Solver::Summary::line_search_type

  Type of the line search algorithm used.

LineSearchInterpolationType Solver::Summary::line_search_interpolation_type

  When performing line search, the degree of the polynomial used to approximate the objective function.

NonlinearConjugateGradientType Solver::Summary::nonlinear_conjugate_gradient_type

  If the line search direction is NONLINEAR_CONJUGATE_GRADIENT, then this indicates the particular variant of non-linear conjugate gradient used.

int Solver::Summary::max_lbfgs_rank

  If the type of the line search direction is LBFGS, then this indicates the rank of the Hessian approximation.