New subspace minimization conjugate gradient methods based on regularization model for unconstrained optimization

TL;DR

This paper introduces two novel subspace minimization conjugate gradient methods based on p-regularization models, demonstrating their global and R-linear convergence, and showing superior performance over existing methods on benchmark problems.

Contribution

The paper proposes new conjugate gradient methods using p-regularization models with a special scaled norm, providing convergence analysis and superior numerical results.

Findings

Methods outperform existing conjugate gradient algorithms on CUTEr benchmark.

Proven global convergence under mild assumptions.

Achieved R-linear convergence rate.

Abstract

In this paper, two new subspace minimization conjugate gradient methods based on $p - $regularization models are proposed, where a special scaled norm in $p - $regularization model is analyzed. Different choices for special scaled norm lead to different solutions to the $p - $regularized subproblem. Based on the analyses of the solutions in a two-dimensional subspace, we derive new directions satisfying the sufficient descent condition. With a modified nonmonotone line search, we establish the global convergence of the proposed methods under mild assumptions. $R - $linear convergence of the proposed methods are also analyzed. Numerical results show that, for the CUTEr library, the proposed methods are superior to four conjugate gradient methods, which were proposed by Hager and Zhang (SIAM J Optim 16(1):170-192, 2005), Dai and Kou (SIAM J Optim 23(1):296-320, 2013), Liu and Liu (J Optim Theory Appl 180(3):879-906, 2019) and Li et al. (Comput Appl Math 38(1): 2019), respectively.