A Solution Concept for Convex Vector Optimization Problems based on a User-defined Region of Interest

TL;DR

This paper introduces a new solution concept for convex vector optimization problems that uses homogenization and relative error measures, supporting user-defined regions of interest and avoiding common structural assumptions.

Contribution

It proposes a robust, scalable solution approach based on homogenization that handles unbounded problems and incorporates user-defined regions of interest without requiring polyhedral assumptions.

Findings

The method employs a single precision parameter for simplicity.

It remains robust under scaling due to relative error measures.

Supports iterative refinement within user-defined regions of interest.

Abstract

This work addresses arbitrary convex vector optimization problems, which constitute a general framework for multi-criteria decision-making in diverse real-world applications. Due to their complexity, such problems are typically tackled using polyhedral approximation. Existing solution concepts rely on additional assumptions, such as boundedness, polyhedrality of the ordering cone, or existence of interior points in the ordering cone, and typically focus on absolute error measures. We introduce a solution concept based on the homogenization of the upper image that employs relative error measures and avoids additional structural assumptions. Although minimality is not explicitly required, a form of approximate minimality is implicitly ensured. The concept is straightforward, requiring only a single precision parameter and, owing to relative errors, remains robust under scaling of the target functions. Homogenization also eliminates the need for the binary distinction between points far from the origin and directions which can lead to numerical difficulties. Furthermore, in practice decision-makers often identify a region where preferred solutions are expected. Our concept supports both a global overview of the upper image and a refined local perspective within such a user-defined region of interest (RoI).We present a decision-making procedure enabling iterative refinement of this region and the associated preferences.