The Multi-Objective Polynomial Optimization

TL;DR

This paper explores polynomial multi-objective optimization, providing convex geometric insights, methods for solving scalarization problems, and techniques to verify Pareto points and their existence.

Contribution

It introduces convex representations for Pareto values and develops relaxations and checks for scalarization problems and Pareto point existence in polynomial optimization.

Findings

Convex geometric characterization of Pareto values

Relaxation methods for scalarization problems

Procedures to verify Pareto points and detect unboundedness

Abstract

The multi-objective optimization is to optimize several objective functions over a common feasible set. Since the objectives usually do not share a common optimizer, people often consider (weakly) Pareto points. This paper studies multi-objective optimization problems that are given by polynomial functions. First, we study the convex geometry for (weakly) Pareto values and give a convex representation for them. Linear scalarization problems (LSPs) and Chebyshev scalarization problems (CSPs) are typical approaches for getting (weakly) Pareto points. For LSPs, we show how to use tight relaxations to solve them, how to detect existence or nonexistence of proper weights. For CSPs, we show how to solve them by moment relaxations. Moreover, we show how to check if a given point is a (weakly) Pareto point or not and how to detect existence or nonexistence of (weakly) Pareto points. We also study how to detect unboundedness of polynomial optimization, which is used to detect nonexistence of proper weights or (weakly) Pareto points.