A barrier-type method for multiobjective optimization

TL;DR

This paper introduces a multiobjective barrier method (MBM) for solving constrained multicriteria optimization problems, extending classical scalar methods with new auxiliary functions and convergence guarantees to Pareto optima.

Contribution

It develops a novel multiobjective barrier method using auxiliary monotonic functions, providing convergence analysis and an implementable version for local optima.

Findings

Convergence to Pareto and weak Pareto optima under mild assumptions.

A flexible auxiliary function framework for multiobjective barrier methods.

An implementable MBM version with proven convergence properties.

Abstract

For solving constrained multicriteria problems, we introduce the multiobjective barrier method (MBM), which extends the scalar-valued internal penalty method. This multiobjective version of the classical method also requires a penalty barrier for the feasible set and a sequence of nonnegative penalty parameters. Differently from the single-valued procedure, MBM is implemented by means of an auxiliary "monotonic" real-valued mapping, which may be chosen in a quite large set of functions. Here, we consider problems with continuous objective functions, where the feasible sets are defined by finitely many continuous inequalities. Under mild assumptions, and depending on the monotonicity type of the auxiliary function, we establish convergence to Pareto or weak Pareto optima. Finally, we also propose an implementable version of MBM for seeking local optima and analyze its convergence to Pareto or weak Pareto solutions.