Pattern Search Algorithms for Mixed Variable Programming

Abstract

A new class of algorithms for solving nonlinearly constrained mixed variable optimization problems is presented. This class combines and extends the Audet-Dennis generalized pattern search (GPS) algorithms for bound constrained mixed variable optimization, and their GPS-filter algorithms for general nonlinear constraints. In generalizing existing algorithms, new theoretical convergence results are presented that reduce seamlessly to existing results for more specific classes of problems. While no local continuity or smoothness assumptions are required to apply the algorithm, a hierarchy of theoretical convergence results based on the Clarke calculus is given, in which local smoothness dictates what can be proved about certain limit points generated by the algorithm. We believe this is the first algorithm with provable convergence results to directly target this class of problems.

Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO

“…Local-search CDFO contributions have been extended for handling mixed-integer variable problems in the work of (1) Audet and Dennis (2001) for box-constrained problems; (2) Abramson et al (2001) for general constrained CDFO problems through a filter approach; (3) Lucidi et al (2005) and Abramson et al (2009a) for constrained problems with known linear constraints; (4) Liuzzi et al (2012) for box-constrained problems based on the earlier method desribed in (Lucidi and Sciandrone, 2002); (5) Liuzzi et al (2015b) for general constrained mixed-integer problems extending an earlier sequential penalty approach described in (Liuzzi et al, 2010). A contribution which diverges from the above, but belongs in the local-search direct-search category is developed for purely discrete problems, for which an implicit and dense closed set is available (Vicente, 2009).…”

Using pattern search methods for surface structure determination of nanomaterials

“…For details about the GPS method for constrained optimization, see sections 7 and 8 of [21]. The GPS method for mixed variable problems is described in more detail in [25].…”

Derivative-free methods for bound constrained mixed-integer optimization

“…In [1,2,5,8] a problem more general than (1) has been considered by allowing also for the presence of categorical variables. The algorithms proposed in the papers [1,2,5,8] are based on the idea of alternating between a local minimization with respect to the continuous variables and a local search with respect to the discrete variables.…”

“…The algorithms proposed in the papers [1,2,5,8] are based on the idea of alternating between a local minimization with respect to the continuous variables and a local search with respect to the discrete variables. The common feature of the methods is represented by the fact that the discrete neighborhood structure (that is needed to define the local search) is fixed a priori at every iterate.…”