Norm minimization-based convex vector optimization algorithms

Abstract

This thesis is concerned with convex vector optimization problems (CVOP). We propose an outer approximation algorithm (Algorithm 1) for solving CVOPs. In each iteration, the algorithm solves a norm-minimizing scalarization for a reference point in the objective space. The idea is inspired by some Benson-type algorithms in the literature that are based on Pascoletti-Serafini scalarization. Since this scalarization needs a direction parameter, the efficiency of these algorithms depend on the selection of the direction parameter. In contrast, our algorithm is free of direction biasedness since it solves a scalarization that is based on minimizing a norm. However, the structure of such algorithms, including ours, has some built-in limitation which makes it difficult to perform convergence analysis. To overcome this, we modify the algorithm by introducing a suitable compact subset of the upper image. After the modification, we have Algorithm 2 in which norm-minimizing scalarizations are solved for points in the compact set. To the best of our knowledge, Algorithm 2 is the first algorithm for CVOPs, which is proven to be finite. Finally, we propose a third algorithm for the purposes of con-vergence analysis (Algorithm 3), where a modified norm-minimizing scalarization is solved in each iteration. This scalarization includes an additional constraint which ensures that the algorithm deals with only a compact subset of the upper image from the beginning. Besides having the finiteness result, Algorithm 3 is the first CVOP algorithm with an estimate of a convergence rate. The experimental results, obtained using some benchmark test problems, show comparable performance of our algorithms with respect to an existing CVOP algorithm based on Pascoletti-Serafini scalarization.