Geometric duality results and approximation algorithms for convex vector optimization problems

TL;DR

This paper introduces a new geometric duality framework for convex vector optimization problems, establishing a duality without fixed directions and proposing algorithms that solve primal and dual problems simultaneously, with demonstrated effectiveness on test instances.

Contribution

It develops a direction-free geometric duality approach and algorithms for convex vector optimization, improving upon existing methods by avoiding direction bias and enabling simultaneous primal-dual solutions.

Findings

The dual problem's image is a convex cone independent of direction.

A polyhedral approximation of one image yields an approximation of the other.

Algorithms perform well on randomly generated instances using primal error and hypervolume metrics.

Abstract

We study geometric duality for convex vector optimization problems. For a primal problem with a $q$-dimensional objective space, we formulate a dual problem with a $(q+1)$-dimensional objective space. Consequently, different from an existing approach, the geometric dual problem does not depend on a fixed direction parameter and the resulting dual image is a convex cone. We prove a one-to-one correspondence between certain faces of the primal and dual images. In addition, we show that a polyhedral approximation for one image gives rise to a polyhedral approximation for the other. Based on this, we propose a geometric dual algorithm which solves the primal and dual problems simultaneously and is free of direction-biasedness. We also modify an existing direction-free primal algorithm in a way that it solves the dual problem as well. We test the performance of the algorithms for randomly generated problem instances by using the so-called primal error and hypervolume indicator as performance measures.