$\gamma$-Competitiveness: An Approach to Multi-Objective Optimization with High Computation Costs in Lipschitz Functions

TL;DR

This paper introduces a hyperparameter-free, interpretable scalarization method for multi-objective optimization with high computational costs, especially effective for Lipschitz functions with limited evaluations, demonstrated on flow problems.

Contribution

It proposes the SWCM and CAoLF methods, extending competitive solutions to reduce computational costs in multi-objective optimization involving Lipschitz functions.

Findings

Efficiently solves MOO problems with limited function evaluations.

Eliminates hyperparameter tuning in scalarization methods.

Demonstrates scalability on flow network problems.

Abstract

In practical engineering and optimization, solving multi-objective optimization (MOO) problems typically involves scalarization methods that convert a multi-objective problem into a single-objective one. While effective, these methods often incur significant computational costs due to iterative calculations and are further complicated by the need for hyperparameter tuning. In this paper, we introduce an extension of the concept of competitive solutions and propose the Scalarization With Competitiveness Method (SWCM) for multi-criteria problems. This method is highly interpretable and eliminates the need for hyperparameter tuning. Additionally, we offer a solution for cases where the objective functions are Lipschitz continuous and can only be computed once, termed Competitiveness Approximation on Lipschitz Functions (CAoLF). This approach is particularly useful when computational resources are limited or re-computation is not feasible. Through computational experiments on the minimum-cost concurrent flow problem, we demonstrate the efficiency and scalability of the proposed method, underscoring its potential for addressing computational challenges in MOO across various applications.