Few for Many: Tchebycheff Set Scalarization for Many-Objective Optimization

TL;DR

This paper introduces a Tchebycheff set scalarization method that efficiently finds a small set of representative solutions to cover many objectives in multi-objective optimization, addressing scalability issues.

Contribution

The paper proposes a novel scalarization approach that finds few solutions to represent large sets of objectives, improving scalability for many-objective problems.

Findings

Effective in covering over 100 objectives with only 5 solutions

Outperforms existing methods in scalability and solution quality

Provides theoretical guarantees for the optimization approach

Abstract

Multi-objective optimization can be found in many real-world applications where some conflicting objectives can not be optimized by a single solution. Existing optimization methods often focus on finding a set of Pareto solutions with different optimal trade-offs among the objectives. However, the required number of solutions to well approximate the whole Pareto optimal set could be exponentially large with respect to the number of objectives, which makes these methods unsuitable for handling many optimization objectives. In this work, instead of finding a dense set of Pareto solutions, we propose a novel Tchebycheff set scalarization method to find a few representative solutions (e.g., 5) to cover a large number of objectives (e.g., $>100$) in a collaborative and complementary manner. In this way, each objective can be well addressed by at least one solution in the small solution set. In addition, we further develop a smooth Tchebycheff set scalarization approach for efficient optimization with good theoretical guarantees. Experimental studies on different problems with many optimization objectives demonstrate the effectiveness of our proposed method.