New merit functions for multiobjective optimization and their properties

TL;DR

This paper introduces new merit functions for multiobjective optimization that possess desirable mathematical properties, facilitating more effective solution estimation and problem-solving.

Contribution

The paper proposes novel merit functions tailored for multiobjective optimization with various objective types, demonstrating their favorable properties under reasonable assumptions.

Findings

Merit functions are zero at solutions and positive elsewhere.

Proposed functions are lower semicontinuous, convex, and composite objectives compatible.

Functions exhibit properties like ease of computation, continuity, and bounded error.

Abstract

A merit (gap) function is a map that returns zero at the solutions of problems and strictly positive values otherwise. Its minimization is equivalent to the original problem by definition, and it can estimate the distance between a given point and the solution set. Ideally, this function should have some properties, including the ease of computation, continuity, differentiability, boundedness of the level set, and error boundedness. In this work, we propose new merit functions for multiobjective optimization with lower semicontinuous objectives, convex objectives, and composite objectives, and we show that they have such desirable properties under reasonable assumptions.