Primal and dual approximation algorithms for convex vector optimization problems

Abstract

Two approximation algorithms for solving convex vector optimization problems (CVOPs) are provided. Both algorithms solve the CVOP and its geometric dual problem simultaneously. The first algorithm is an extension of Benson's outer approximation algorithm, and the second one is a dual variant of it. Both algorithms provide an inner as well as an outer approximation of the (upper and lower) images. Only one scalar convex program has to be solved in each iteration. We allow objective and constraint functions that are not necessarily differentiable, allow solid pointed polyhedral ordering cones, and relate the approximations to an appropriate -solution concept. Numerical examples are provided.

Tractability of convex vector optimization problems in the sense of polyhedral approximations

“…As discussed in Section 1, Benson-type algorithms proposed in [6,16] are designed to solve bounded CVOPs. Note that a problem is bounded if and only if it is self-bounded and recc P = C. Then, by Proposition 4.12 and Theorem 4.14, a problem is bounded if and only if W = C + .…”

“…For solving convex vector optimization problems, convex upper closed sets with respect to the ordering cone of the problem play an important role as the image of the set of all weak minimizers can be seen as (a subset of) the boundary of a convex upper closed set known as the upper image. Indeed, there are solution concepts for convex vector optimization problems that involve generating (approximations to) the upper image, see, for instance, [15,16]. For linear vector optimization problems, it is possible to generate this set by a finite set of points and a finite set of directions [15], whereas for nonlinear convex vector optimization problems, a solution usually generates an inner and an outer approximation to the upper image [16].…”

“…Note that the image of a (weak) minimizer is a single point on the boundary of the upper image. For bounded convex vector optimization problems, an ǫ-solution concept which generates inner and outer approximations to the whole upper image is given in [16] as follows.…”

“…Assume for now that problem (P) is self-bounded. Clearly, if one can compute the recession cone recc P of the upper image P and if the recession cone is polyhedral and pointed, then it is possible to apply the approximation algorithms for bounded CVOPs in the literature, where the ordering cone is taken as recc P, see [6,16]. Note that, it is in general difficult to check if the recession cone of the upper image is polyhedral and pointed.…”

“…In [16], an ǫ-solution concept is given as a finite set of weak minimizers which generates an inner and an outer approximation to the Pareto frontier. There are Benson-type algorithms that find ǫ-solutions to convex vector optimization problems (CVOPs) assuming that the feasible region is compact, see [6,16]. Note that if the feasible region is compact, then the problem is bounded, that is, the image of the feasible region in the objective space is included in a shifted cone, namely a + K for some a ∈ R q and K being the ordering cone of the problem.…”