Simpat: Self-Bounding Direct Search Method for Optimization

Abstract

The Simplex search method has been modified to include a new procedure for dealing with bounds on the independent variables. Through recursive partitioning of the independent variable set, Pattern search is merged with Simplex search to form a composite hillclimber named Simpat. In Simpat, Pattern search is used for those variables which are at or are very near their bounds, while the Simplex method is applied to the remaining variables. Consequently, Simpat is self-bounding and is a direct search method in that it requires only values for the objective function in order to proceed. Results on test problems have been most encouraging. In conjunction with a penalty function strategy for handling constraints, Simpat has proved extremely successful in optimizing realistic engineering-economic models-including several which were exceedingly large and complex.

Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods

“…Direct search methods are usually presented for the case of unconstrained optimization. What is not so widely appreciated is that since the time they first appeared in the 1950s and 1960s, they have been adapted in various ways to handle constrained problems [39,54,84,95,122,137,139,152,157,178,194,207,227,245].…”

“…In fact, the early developers of direct search methods for constrained optimization were aware that their heuristics did not always work in practice. For instance, see Box's comments on Rosenbrock's method in [39] or Keefer's comments on Box's method in [152]. At the time, the emphasis in optimization was on the development of new computational methods rather than on establishing their theoretical properties.…”

“…The first reference that we can find in the direct search literature which explicitly observes that the coordinate directions should be used as search directions when there are only bound constraints is Keefer's Simpat algorithm [152]. Keefer notes that the pattern associated with the method of Hooke and Jeeves [139] conforms in a natural way to the boundary of the feasible region and is well suited for coping with bounds.…”

“…Applying the version of compass search just discussed to problem (8.1) produces exactly the same iterates as applying compass search method to the unconstrained problem minimizef (x), (8.2) wheref In fact, this is one classical approach used with direct search methods to ensure that the iterates produced remain feasible (see, for instance, [152,194,245], as well as the comments in section 8.3.3). GSS (and other direct search) methods are directly applicable to (8.2) using the exact penalty function (8.3).…”

Hybrid pattern search and simulated annealing for fuzzy production planning problems

“…In fact, the early developers of direct search methods for constrained optimization were aware that their heuristics did not always work in practice. For instance, refer to Box's comments on Rosenbrock's method in [13] or Keefer's comments on Box's method in [14]. At the time, the emphasis in optimization was on the development of new computational methods rather than on establishing their theoretical properties.…”