Derivative-free methods for bound constrained mixed-integer optimization

Abstract

We consider the problem of minimizing a continuously differentiable function of several variables subject to simple bound constraints where some of the variables are restricted to take integer values. We assume that the first order derivatives of the objective function can be neither calculated nor approximated explicitly. This class of mixed integer nonlinear optimization problems arises frequently in many industrial and scientific applications and this motivates the increasing interest in the study of derivative-free methods for their solution. The continuous variables are handled by a linesearch strategy whereas to tackle the discrete ones we employ a local search-type approach. We propose different algorithms which are characterized by the way the current iterate is updated and by the stationarity conditions satisfied by the limit points of the sequences they produce. Key words: Derivative-free optimization, bound constrained optimization, mixed-integer nonlinear programming.3.

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).…”

Derivative-Free Methods for Mixed-Integer Constrained Optimization Problems

“…The proposed method combines two basic ingredients, that are a derivative-free optimization for bound constrained mixed integer problems and a penalty function approach for handling of nonlinear constraints. In particular, integer variables are tackled by a Discrete search procedure which is similar to the one defined in [17]. The presence of nonlinear constraints is accounted for by means of a derivative-free sequential penalty approach like that described and analyzed in [18].…”

“…The main parts of the method are the Continuous search and Discrete search procedures. The Continuous search and Discrete search procedures, which investigate the corresponding coordinate direction, are similar to those described in [17], but they are applied to the penalty function P (x; ǫ). At the end of every main iteration the algorithm computes the new values both for the penalty parameter and the sufficient decrease parameter, which are fundamental ingredients in the MINLP case as they allow us to guarantee the convergence of the proposed algorithm.…”

“…Finally, in the recent paper [14] a scatter search procedure is proposed to solve black-box optimization problems where all of the variables can only assume integer values. In this paper we extend the approach proposed in [17] for box constrained mixed integer problems by using a sequential quadratic penalty approach described and analyzed in [18]. The presence of both integer variables and nonlinear constraints makes the extension of the approaches proposed in [17] not straightforward.…”

“…In this paper we extend the approach proposed in [17] for box constrained mixed integer problems by using a sequential quadratic penalty approach described and analyzed in [18]. The presence of both integer variables and nonlinear constraints makes the extension of the approaches proposed in [17] not straightforward. In particular, the possible alternation of minimizations of continuous and discrete variables needs new theoretical analysis of the algorithms.…”