Convergence analysis of a norm minimization-based convex vector optimization algorithm

TL;DR

This paper introduces a finite-iteration outer approximation algorithm for bounded convex vector optimization problems, demonstrating convergence rates that depend on the norm used, with improvements for Euclidean norms.

Contribution

It develops a new outer approximation algorithm based on a modified norm-minimizing scalarization, providing convergence guarantees and rates for CVOPs.

Findings

Algorithm terminates finitely with a polyhedral outer approximation within tolerance.

Convergence rate is (k^{1/(1-q)}) for general norms.

Improved convergence rate (k^{2/(1-q)}) for Euclidean norm.

Abstract

In this work, we propose an outer approximation algorithm for solving bounded convex vector optimization problems (CVOPs). The scalarization model solved iteratively within the algorithm is a modification of the norm-minimizing scalarization proposed in Ararat et al. (2022). For a predetermined tolerance $\epsilon>0$, we prove that the algorithm terminates after finitely many iterations, and it returns a polyhedral outer approximation to the upper image of the CVOP such that the Hausdorff distance between the two is less than $\epsilon$. We show that for an arbitrary norm used in the scalarization models, the approximation error after $k$ iterations decreases by the order of $\mathcal{O}(k^{{1}/{(1-q)}})$, where $q$ is the dimension of the objective space. An improved convergence rate of $\mathcal{O}(k^{{2}/{(1-q)}})$ is proved for the special case of using the Euclidean norm.