Logarithmic Quasi-distance Proximal Point Scalarization Method for Multi-Objective Programming

TL;DR

This paper introduces a novel scalarization method for multi-objective optimization using logarithmic quasi-distance regularization, demonstrating convergence properties and identifying weak Pareto solutions.

Contribution

It proposes a variation of the proximal point scalarization method employing logarithm and quasi-distance regularization, despite losing convexity and differentiability, and proves convergence to weak Pareto solutions.

Findings

Sequence x^k is bounded

Sequence z^k converges

Accumulation points are weak Pareto solutions

Abstract

Recently, Greg\'orio and Oliveira developed a proximal point scalarization method (applied to multi-objective optimization problems) for an abstract strict scalar representation with a variant of the logarithmic-quadratic function of Auslender et al. as regularization. In this study, a variation of this method is proposed, using the regularization with logarithm and quasi-distance, which entails losing important properties, such as convexity and differentiability. However, proceeding differently, it is shown that any sequence \{(x^k, z^k)\} \includ R^n \times R^{m}_{++} generated by the method satisfies: \{z^k\} is convergent and \{x^k\} is bounded and its accumulation points are weak pareto solutions of the unconstrained multi-objective optimization problem