Benson type algorithms for linear vector optimization and applications

TL;DR

This paper introduces improved Benson type algorithms for linear vector optimization that reduce computational effort and extend applicability to more general problem settings, with practical numerical examples including risk measures.

Contribution

It presents new primal and dual Benson algorithms solving only one scalar LP per iteration and extends the methods to arbitrary pointed solid polyhedral cones.

Findings

Algorithms require fewer LP solves per iteration

Extended methods apply to broader cone types

Numerical examples demonstrate practical effectiveness

Abstract

New versions and extensions of Benson's outer approximation algorithm for solving linear vector optimization problems are presented. Primal and dual variants are provided in which only one scalar linear program has to be solved in each iteration rather than two or three as in previous versions. Extensions are given to problems with arbitrary pointed solid polyhedral ordering cones. Numerical examples are provided, one of them involving a new set-valued risk measure for multivariate positions.