/** \page ConstrainedProblems Constrained minimization Consider the general nonlinear programming problem minimize \f[ f(x) \f] subject to \f[ h(x) = 0, \f] \f[ g(x) \ge 0. \f]

In this section, we provide the framework for the general nonlinear programming problem. We also examine the construction of linear, nonlinear and compound constraints.

- \ref LinearConstraints
- \ref NonLinearConstraints
- \ref CompoundConstraintsDoc

If a constraint has finite lower and upper bounds, OPT++ treats it as two separate constraints. In the computation of the residuals and gradients, constraints with finite lower bounds appear first followed by those with finte upper bounds. Consequently, in the optimization summary, you will see the constraint count is double the original number of constraints.

Now, we have all the necessary tools to build a nonlinear programming problem.

- \ref Problem
- \ref Fragments

\section Problem Creating a general nonlinear programming problem Let's consider Problem 65 from the Hock and Schittkowski suite of test problems: minimize \f[ (x_1 - x_2)^2 + (1/9)(x_1 + x_2 - 10)^2 + (x_3 - 5)^2 \f] subject to \f[ x_1^2 + x_2^2 + x_3^2 \le 48, \f] \f[-4.5 \le x_1 \le 4.5, \f] \f[-4.5 \le x_2 \le 4.5, \f] \f[ -5.0 \le x_3 \le 5.0 \f] Step 1: Build your constraints. The constraint set consists of bounds on the variables plus one nonlinear inequality. \code // Number of bounds and number of nonlinear constraints int numBds = 3, ncnln = 1; ColumnVector lower(numBds), upper(numBds); // Construct the nonlinear equation NLP* chs65 = new NLP( new NLF2(numBds,ncnln,ineq_hs65,init_hs65) ); Constraint ineq = new NonLinearInequality(chs65); // Construct the bound constraints lower << -4.5 << -4.5 << -5.0; upper << 4.5 << 4.5 << 5.0 ; Constraint bc = new BoundConstraint(n,lower,upper); // Construct the compound constraint CompoundConstraint* constraints = new CompoundConstraint(ineq,bc); \endcode Step 2: Create a constrained problem where the objective function has analytic Hessians. \code NLF2 hs65_problem(n,hs65,init_hs65,constraints); \endcode \section Fragments Specifying the optimization method In OPT++, the only freely available method to solve a general nonlinear problem is OptNIPS, a nonlinear interior-point method. However, we do provide an wrapper to NPSOL, a sequential quadratic programming algorithm. For more information, go to the Next Section: Parallel optimization | Back to Solvers Page Last revised July 13, 2006 */