An Algorithm to Solve Polyhedral Convex Set Optimization Problems

TL;DR

This paper presents an algorithm for solving polyhedral convex set optimization problems by combining linear vector optimization and vertex enumeration to ensure solutions meet both infimum attainment and minimality criteria.

Contribution

It introduces a two-phase algorithm that efficiently computes solutions for set optimization problems with polyhedral convex objectives, integrating linear programming and vertex enumeration.

Findings

The algorithm successfully computes solutions satisfying both infimum and minimality.

It leverages linear inequalities to define the objective map's graph.

The method is applicable to problems with polyhedral convex set objectives.

Abstract

An algorithm which computes a solution of a set optimization problem is provided. The graph of the objective map is assumed to be given by finitely many linear inequalities. A solution is understood to be a set of points in the domain satisfying two conditions: the attainment of the infimum and minimality with respect to a set relation. In the first phase of the algorithm, a linear vector optimization problem, called the vectorial relaxation, is solved. The resulting pre-solution yields the attainment of the infimum but, in general, not minimality. In the second phase of the algorithm, minimality is established by solving certain linear programs in combination with vertex enumeration of some values of the objective map.