Outer approximation algorithms for convex vector optimization problems

TL;DR

This paper introduces a flexible outer approximation framework for convex vector optimization that employs iterative Pascoletti-Serafini scalarization, with novel parameter selection methods that improve computational efficiency.

Contribution

It proposes new parameter selection strategies for PS scalarization in outer approximation algorithms, enhancing performance without extra optimization problems.

Findings

The proposed methods reduce runtime significantly.

They achieve comparable or better approximation accuracy.

The algorithms perform well across various test problems.

Abstract

In this study, we present a general framework of outer approximation algorithms to solve convex vector optimization problems, in which the Pascoletti-Serafini (PS) scalarization is solved iteratively. This scalarization finds the minimum 'distance' from a reference point, which is usually taken as a vertex of the current outer approximation, to the upper image through a given direction. We propose efficient methods to select the parameters (the reference point and direction vector) of the PS scalarization and analyze the effects of these on the overall performance of the algorithm. Different from the existing vertex selection rules from the literature, the proposed methods do not require solving additional single-objective optimization problems. Using some test problems, we conduct an extensive computational study where three different measures are set as the stopping criteria: the approximation error, the runtime, and the cardinality of solution set. We observe that the proposed variants have satisfactory results especially in terms of runtime compared to the existing variants from the literature.