A derivative-free descent method in set optimization

Abstract

Based on a vectorization result in set optimization with respect to the set less order relation, this paper shows how to relate two nonempty sets on a computer. This result is developed for generalized convex sets and polyhedral sets in finite dimensional spaces. Using this approach a numerical method for the determination of optimal scenarios is presented. A new derivative-free descent method for the solution of set optimization problems is given together with numerical results in low dimensions.

An algorithmic approach to multiobjective optimization with decision uncertainty

“…As one can see from the definition, one has to be able to verify whether it holds A ⊆ B − R m + for two sets A, B ⊆ R m numerically. In case of polyhedral sets this can be done by using a finite number of support functionals, see [15]. In case of arbitrary closed convex sets one might need an infinite number of such linear functionals.…”

“…For unconstrained set-valued optimization and a similar order relation Löhne and Schrage introduced an algorithm in [26], which is applicable for linear problems only. Jahn presented some derivative-free algorithms, see [15] to find one single solution of the whole set of optimal solutions in case the sets which have to be compared are convex. Köbis and Köbis extended the method from [15] to the nonconvex case, i.e.…”

“…Jahn presented some derivative-free algorithms, see [15] to find one single solution of the whole set of optimal solutions in case the sets which have to be compared are convex. Köbis and Köbis extended the method from [15] to the nonconvex case, i.e. when the sets are nonconvex sets, see [18].…”

“…However, all methods aim at finding one single minimal solution only. In [13] for the first time a method for nonconvex sets is presented, which uses discretization to compare sets and which can find many minimal solutions, but still not all. The procedure was parallelized and implemented on a CPU and GPU.…”

Computing minimal elements of finite families of sets w.r.t. preorder relations in set optimization

“…Set optimization has become an active field of research due to its wide applications (see, e.g., Khan, Tammer and Zȃlinescu [13] and the references therein). Deriving efficient methods for solving set optimization problems is particularly important, and so far several interesting methods have been developed for computing the minimal elements or some approximations of minimal elements (see, e.g., Jahn [10], and Köbis and Köbis [15]).…”