Abstract

Three-term conjugate gradient methods have attracted much attention for large-scale unconstrained problems in recent years, since they have attractive practical factors such as simple computation, low memory requirement, better descent property and strong global convergence property. In this paper, a hybrid three-term conjugate gradient algorithm is proposed and it owns a sufficient descent property, independent of any line search technique. Under some mild conditions, the proposed method is globally convergent for uniformly convex objective functions. Meanwhile, by using the modified secant equation, the proposed method is also global convergence without convexity assumption on the objective function. Numerical results also indicate that the proposed algorithm is more efficient and reliable than the other methods for the testing problems.

Highlights

  • In this paper, we consider the following unconstrained problem: 2021, 9, 1353. https://doi.org/10.3390/math9121353 min f ( x ), x ∈ RnAcademic Editor: Ioannis K

  • Based on the above analyses, we state the steps of our algorithm as follows: Algorithm 1: New hybrid three-term conjugate gradient method (HTTCG)

  • We show that Algorithm 1 owns the sufficient descent property independent of any line search technique

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Summary

Introduction

We consider the following unconstrained problem: 2021, 9, 1353. https://doi.org/. Note that the HS method automatically satisfies the conjugate condition dkT+1 yk = 0, independently of any line search technique. The new method owns the sufficient descent property independent of the accuracy of the line search technique. The global convergence is established for uniformly convex objective functions. For general functions without convexity assumption, the global convergence is established by using the modified secant condition in [32].

Motivation and Algorithm
Algorithm for Uniformly Convex Functions
Convergence for Uniformly Convex Functions
Convergence for General Nonlinear Functions
Numerical Results
Numerical Performance of Algorithm 2
A Quadratic Function QF2
Accelerated Strategy for Algorithm 2
Conclusions

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