Blurb::
Pattern search, derivative free optimization method
Description::
Pattern search techniques are nongradient-based optimization methods
which use a set of offsets from the current iterate to locate improved
points in the design space.


*See the page :ref:`topic-package_scolib` for important information regarding all SCOLIB methods*

``coliny_pattern_search`` supports concurrency up to the size of
the search pattern

Traditional pattern search methods search with a fixed pattern of
search directions to try to find improvements to the current iterate.
The SCOLIB pattern search methods generalize this simple algorithmic
strategy to enable control of how the search pattern is adapted, as
well as how each search pattern is evaluated. The ``stochastic`` and
``synchronization`` specifications denote how the the trial points are
evaluated. The ``stochastic`` specification indicates that the trial
points are considered in a random order. For parallel pattern search,
``synchronization`` dictates whether the evaluations are scheduled
using a ``blocking`` scheduler or a ``nonblocking`` scheduler.  In the
``blocking`` case, all points in the pattern are evaluated (in
parallel), and if the best of these trial points is an improving
point, then it becomes the next iterate. These runs are reproducible,
assuming use of the same seed in the ``stochastic`` case.  In the
``nonblocking`` case, all points in the pattern may not be evaluated,
since the first improving point found becomes the next iterate. Since
the algorithm steps will be subject to parallel timing variabilities,
these runs will not generally be repeatable. The ``synchronization``
specification has similar connotations for sequential pattern
search. If ``blocking`` is specified, then each sequential iteration
terminates after all trial points have been considered, and if
``nonblocking`` is specified, then each sequential iteration terminates
after the first improving trial point is evaluated. In this release,
both ``blocking`` and ``nonblocking`` specifications result in blocking
behavior (except in the case where ``exporatory_moves`` below is set to
``adaptive_pattern``). Nonblocking behavior will be re-enabled after
some underlying technical issues have been resolved.

The particular form of the search pattern is controlled by the
``pattern_basis`` specification. If ``pattern_basis`` is ``coordinate``
basis, then the pattern search uses a plus and minus offset in each
coordinate direction, for a total of *2n* function evaluations in the
pattern. This case is depicted in Figure 5.3 for three coordinate
dimensions.

\image html pattern_search.jpg "Figure 5.3 Depiction of coordinate pattern search algorithm"
\image latex pattern_search.eps "Depiction of coordinate pattern search algorithm" width=10cm

If ``pattern_basis`` is ``simplex``, then pattern search uses a minimal
positive basis simplex for the parameter space, for a total of *n+1*
function evaluations in the pattern. Note that the ``simplex`` pattern
basis can be used for unbounded problems only. The
``total_pattern_size`` specification can be used to augment the basic
``coordinate`` and ``simplex`` patterns with additional function
evaluations, and is particularly useful for parallel load balancing.
For example, if some function evaluations in the pattern are dropped
due to duplication or bound constraint interaction, then the
``total_pattern_size`` specification instructs the algorithm to generate
new offsets to bring the total number of evaluations up to this
consistent total.

The ``exploratory_moves`` specification controls how the search pattern
is adapted. (The search pattern can be adapted after an improving
trial point is found, or after all trial points in a search pattern
have been found to be unimproving points.)  The following exploratory
moves selections are supported by SCOLIB:


- The ``basic_pattern`` case is the simple pattern search approach, which uses the same pattern in each iteration.
- The ``multi_step`` case examines each trial step in the pattern in turn. If a successful step is found, the pattern search continues examining trial steps about this new point. In this manner, the effects of multiple successful steps are cumulative within a single iteration. This option does not support any parallelism and will result in a serial pattern search.
- The ``adaptive_pattern`` case invokes a pattern search technique that adaptively rescales the different search directions to maximize the number of redundant function evaluations. See :cite:p:`Hart2001c` for details of this method.  In preliminary experiments, this method had more robust performance than the standard ``basic_pattern`` case in serial tests.  This option supports a limited degree of parallelism. After successful iterations (where the step length is not contracted), a parallel search will be performed. After unsuccessful iterations (where the step length is contracted), only a single evaluation is performed.

The ``initial_delta`` and ``variable_tolerance`` specifications provide the
initial offset size and the threshold size at which to terminate the
algorithm. For any dimension that has both upper and lower bounds,
this step length will be internally rescaled to provide search steps
of length ``initial_delta`` *range* 0.1. This rescaling does not
occur for other dimensions, so search steps in those directions have
length ``initial_delta``. Note that the factor of 0.1 in the rescaling
could result in an undesirably small initial step. This can be offset
by providing a large ``initial_delta``.

In general, pattern search methods can expand and contract their step
lengths. SCOLIB pattern search methods contract the step length by the
value ``contraction_factor``, and they expand the step length by the
value (1/contraction_factor). The ``expand_after_success`` control
specifies how many successful objective function improvements must
occur with a specific step length prior to expansion of the step
length, whereas the ``no_expansion`` flag instructs the algorithm to
forgo pattern expansion altogether.

Finally, constraint infeasibility can be managed in a somewhat more
sophisticated manner than the simple weighted penalty function. If the
``constant_penalty`` specification is used, then the simple weighted
penalty scheme described above is used. Otherwise, the constraint
penalty is adapted to the value ``constraint_penalty/L``, where L is
the the smallest step length used so far.


*Expected HDF5 Output*

If Dakota was built with HDF5 support and run with the
:dakkw:`environment-results_output-hdf5` keyword, this method
writes the following results to HDF5:


- :ref:`hdf5_results-best_params`
- :ref:`hdf5_results-best_obj_fncs` (when :dakkw:`responses-objective_functions`) are specified)
- :ref:`hdf5_results-best_constraints`
- :ref:`hdf5_results-calibration` (when :dakkw:`responses-calibration_terms` are specified)
Topics::
package_scolib, package_coliny, global_optimization_methods
Examples::

Theory::

Faq::

See_Also::
