Primal and Dual Approximation Algorithms for Convex Vector Optimization Problems

TL;DR

This paper introduces two approximation algorithms for convex vector optimization problems that efficiently generate inner and outer approximations of the solution set, extending Benson's method to handle non-differentiable functions and dual problems.

Contribution

The paper develops primal and dual approximation algorithms for CVOPs that solve the problem and its dual simultaneously, with convergence guarantees and applicability to non-differentiable functions.

Findings

Algorithms produce inner and outer approximations of the solution set.

Only one scalar convex program is needed per iteration.

Numerical examples demonstrate effectiveness and flexibility.

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 \epsilon-solution concept. Numerical examples are provided.