A Decision-Making Method in Polyhedral Convex Set Optimization

TL;DR

This paper introduces a practical decision-making method for set-valued optimization problems within polyhedral convex sets, enabling the selection of an optimizer through iterative design and visualization, with proven finite convergence.

Contribution

It proposes a novel trial-and-error approach for selecting optimizers in set-valued convex optimization, supported by implementation and finite convergence proof.

Findings

The method allows finite-step identification of an optimizer.

Visualization aids decision-making in set-valued optimization.

Application demonstrated on a bi-objective network flow problem.

Abstract

Optimization problems with set-valued objective functions arise in contexts such as multi-stage optimization with vector-valued objectives. The aim is to identify an optimizer -- a feasible point with an optimal objective value -- based on an ordering relation on a family of sets. When faced with multiple optimizers, a decision maker must choose one. Visualizing the values associated with these optimizers could provide a solid basis for decision-making. However, these values are sets, making it challenging to visualize many of them. Therefore, we propose a method where an optimizer is selected by designing the respective outcome set through a trial-and-error process. In a polyhedral convex setting, we discuss an implementation and prove that an optimizer can be found using this method after a finite number of design steps. We motivate the problem setting and illustrate the process using an example: a two-stage bi-objective network flow problem.