Cubic Regularization Technique of the Newton Method for Vector Optimization

TL;DR

This paper introduces a cubic regularization of the Newton method for vector optimization that guarantees convergence to weakly efficient points without requiring convexity, and demonstrates its effectiveness through theoretical analysis and numerical comparisons.

Contribution

It proposes a novel cubic regularization approach for vector optimization that ensures global convergence and retains local quadratic convergence without line search.

Findings

Achieves $O(k^{-2/3})$ global convergence rate.

Retains local q-quadratic convergence.

Outperforms existing methods on test examples.

Abstract

This study proposes a cubic regularization of the Newton method for generating weakly efficient points of unconstrained vector optimization problems under no convexity assumption on the objective function. It is observed that at a given iterate, the cubic regularized Newton direction is not necessarily a descent direction. In generating the sequence of iterates, no line search is utilized to find a suitable step length to move along the cubic regularized Newton direction. Yet, the proposed method exhibits a global convergence property with $O(k^{-2/3})$ rate of convergence. Further, the local q-quadratic convergence of the Newton method is also retained in the cubic regularization. A new stopping condition is used, which enforces the proposed method to enter in close neighborhood of non-weakly efficient points that are stationary. Thus, the studied technique ends up generating weakly efficient points, not just Pareto critical points. In addition, conditions on the choice of regularization parameter value under which the full cubic regularized Newton step becomes descent are derived. Performance profiles and comparison of the derived method with the existing methods on several test examples are also provided.