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Cubic Regularization Technique of the Newton Method for Vector Optimization

Debdas Ghosh

TL;DR

The paper develops a cubic-regularized Newton method for unconstrained vector optimization with possibly non-convex objectives and Lipschitz Hessians. By formulating a cubic-regularized local model and a computable Newton direction $d_M(x)$, it achieves global convergence at rate $O(k^{-2/3})$ and retains local $q$-quadratic convergence, while guaranteeing convergence to weakly efficient points via a novel stopping criterion $ ho_M$. The method avoids line searches by using an auxiliary function $h_M$ and a practical scheme to select the regularization parameter $M_k$, ensuring descent and facilitating strong convergence properties. Numerical experiments demonstrate competitive performance against existing gradient-based, quasi-Newton, and trust-region methods, highlighting CN’s ability to reach weakly efficient points rather than merely stationary ones. The work establishes a solid framework for parameter-free, robust vector optimization with theoretical guarantees and practical efficacy.

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.

Cubic Regularization Technique of the Newton Method for Vector Optimization

TL;DR

The paper develops a cubic-regularized Newton method for unconstrained vector optimization with possibly non-convex objectives and Lipschitz Hessians. By formulating a cubic-regularized local model and a computable Newton direction , it achieves global convergence at rate and retains local -quadratic convergence, while guaranteeing convergence to weakly efficient points via a novel stopping criterion . The method avoids line searches by using an auxiliary function and a practical scheme to select the regularization parameter , ensuring descent and facilitating strong convergence properties. Numerical experiments demonstrate competitive performance against existing gradient-based, quasi-Newton, and trust-region methods, highlighting CN’s ability to reach weakly efficient points rather than merely stationary ones. The work establishes a solid framework for parameter-free, robust vector optimization with theoretical guarantees and practical efficacy.

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 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.
Paper Structure (8 sections, 15 theorems, 115 equations, 6 figures, 3 tables, 1 algorithm)

This paper contains 8 sections, 15 theorems, 115 equations, 6 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature survey of parameter-free methods
  3. Motivation
  4. Work done
  5. Preliminaries
  6. Cubic Regularization
  7. Numerical Experiments
  8. Conclusion and Final Remarks

Key Result

Lemma 2.1

drummond2014quadratically Let $f:S \to \mathbb{R}^p$ be continuously differentiable. If $\bar{x} \in S$ is a weakly efficient point of vop, then $\bar{x}$ is a stationary point of vop.

Figures (6)

  • Figure 1: Objective components and the function $q_M(0.04, d)$ (Remark \ref{['aux1_17_02']})
  • Figure 2: Error graph for the PNR problem
  • Figure 3: Error graph for the SLCDT2 problem
  • Figure 4: Performance profiles
  • Figure 5: Weakly Pareto critical points generated by Algorithm \ref{['algo']} for a few test problems for fifty random initial points
  • ...and 1 more figures

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.4
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • proof
  • ...and 30 more