Pattern search, derivative free optimization method

Topics

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Specification

Alias: none

Argument(s): none

Required/Optional	Dakota Keyword	Dakota Keyword Description
Optional	constant_penalty	Use a simple weighted penalty to manage feasibility
Optional	no_expansion	Don't allow expansion of the search pattern
Optional	expand_after_success	Set the factor by which a search pattern can be expanded
Optional	pattern_basis	Pattern basis selection
Optional	stochastic	Generate trial points in random order
Optional	total_pattern_size	Total number of points in search pattern
Optional	exploratory_moves	Exploratory moves selection
Optional	synchronization	Select how Dakota schedules function evaluations in a pattern search
Optional	contraction_factor	Amount by which step length is rescaled
Optional	constraint_penalty	Multiplier for the penalty function
Optional	initial_delta	Initial step size for non-gradient based optimizers
Optional	threshold_delta	Stopping criteria based on step length or pattern size
Optional	solution_target	Stopping criteria based on objective function value
Optional	seed	Seed of the random number generator
Optional	show_misc_options	Show algorithm parameters not exposed in Dakota input
Optional	misc_options	Set method options not available through Dakota spec
Optional	model_pointer	Identifier for model block to be used by a 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 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.

Figure 5.3 Depiction of coordinate pattern search algorithm

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[44] 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 threshold_delta 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.

Topics

Specification

Description

See Also