A face search heuristic algorithm for optimizing over the efficient set

Abstract

The problem of optimizing a linear function over the efficient set of a multiple objective linear program is an important but difficult problem in multiple criteria decision making. In this article we present a flexible face search heuristic algorithm for the problem. Preliminary computational experiments indicate that the algorithm gives very good estimates of the global optimum with relatively little computational effort. © 1993 John Wiley & Sons, Inc.

Minimal Representations of a Face of a Convex Polyhedron and Some Applications

“…Using a minimal representation of a face to represent it gives us special advantages in many methods, for example, face search methods, face decomposition-based methods, descriptor set-based methods (methods are based on descriptor sets for faces), etc. In solving a problem of optimizing a function over the efficient set of a multiple objective linear programming (MOLP) problem by face search methods or by descriptor set-based methods (see, e.g., [1,13,17]) and in finding the efficient set or determining all maximal efficient faces of an MOLP problem by top-down search methods (see, e.g., [12,19,21]), the number of descriptor sets for faces of the constraint polyhedron that need to be considered can be reduced if the constraint polyhedron of the MOLP problem is represented by a minimal representation of it. In addition, based on [19, Property 2.9] and Properties 6.7-6.8, the method in [19] can be improved on the basis of representing the constraint polyhedron by a minimal representation of it because the maximal descriptor index sets for all descriptor sets whose dimensions are elements of the set {n, (n − 1), (n − 2)} need not be determined for this method, where n is the dimension of the constraint polyhedron (the constraint polyhedron is a special face of it) and the dimension of a descriptor set is one of the face described by it (see [19] or [20] for more details).…”

Generating equidistant representations in biobjective programming

“…Additionally, a similar bilevel formulation has been used in the context of MOLPs in [36][37][38][39][40]. In these papers, an additional linear function (i.e., not necessarily a criterion function of the MOLP) is optimized over the (weakly) efficient solutions of the problem.…”

“…Example 2 [45], shown in (40), is a nonconvex problem with a disconnected Pareto set and a connected weak Pareto set. We first solved the problem with respect to the Note that in both figures we "zoomed in" on the interesting portion of the nondominated set.…”