Convergence Analysis of a Norm Minimization-Based Convex Vector Optimization Algorithm

Abstract

SIAM Journal on Optimization, Volume 34, Issue 3, Page 2700-2728, September 2024.
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 [Ç. Ararat, F. Ulus, and M. Umer, J. Optim. Theory Appl., 194 (2022), pp. 681–712]. For a predetermined tolerance [math], 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 [math]. We show that for an arbitrary norm used in the scalarization models, the approximation error after [math] iterations decreases by the order of [math], where [math] is the dimension of the objective space. An improved convergence rate of [math] is proved for the special case of using the Euclidean norm.