A Three-Term Conjugate Gradient Algorithm with Quadratic Convergence for Unconstrained Optimization Problems

We introduce and investigate proper accelerations of the Dai–Liao (DL) conjugate gradient (CG) family of iterations for solving large-scale unconstrained optimization problems. The improvements are based on appropriate modifications of the CG update parameter in DL conjugate gradient methods. The leading idea is to combine search directions in accelerated gradient descent methods, defined based on the Hessian approximation by an appropriate diagonal matrix in quasi-Newton methods, with search directions in DL-type CG methods. The global convergence of the modified Dai–Liao conjugate gradient method has been proved on the set of uniformly convex functions. The efficiency and robustness of the newly presented methods are confirmed in comparison with similar methods, analyzing numerical results concerning the CPU time, a number of function evaluations, and the number of iterative steps. The proposed method is successfully applied to deal with an optimization problem arising in 2D robotic motion control.

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“…Global convergence of the MSMDL iterations is verifed in Teorem 1. Assumption 2 and the proofs of Lemma 3 can be found in [36][37][38].…”

Section: Convergence Analysismentioning

confidence: 99%

“…If the conditions in Assumption 2 hold, then Assumption 1 is satisfed. Lemma 3 (see [36][37][38]). Under the conditions in Assumption 2, there exist real numbers m, M satisfying…”

mentioning

confidence: 99%

A Modified Dai–Liao Conjugate Gradient Method Based on a Scalar Matrix Approximation of Hessian and Its Application

Ivanov

Milovanović

Stanimirović

et al. 2023

Journal of Mathematics

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“…In this section, we will prove another important condition, called global convergence property. In the following lemma, we review the well-known Zoutendijk condition [15], which plays an important role in the proof of the global convergence analysis of SCG method [16], [17] Lemma [18]: Let assumption (A) holds. Suppose any iteration method 2 and 3, and α k is obtained by the SWP.…”

Section: The Global Convergence Propertymentioning

confidence: 99%

A New Paired Spectral Gradient Method to Improve Unconstrained and Non-Linear Optimization

Aziz

Abdullah

2023

KUJSS

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The conjugated spectral gradient (SCG) method is an effective method for non-constrained large-scale nonlinear optimization. In this work, a new spectral conjugate gradient method is proposed with a strong Wolfe-Powell line search (SWP). The new proposal is based on using the formula obtained by comparing the proposed algorithm with previously published conjugate gradient algorithms. Under the usual assumptions, the descent properties and overall global convergence of the proposed method are proved. The proposed method is numerically proven to be effective.

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“…Next, in [52], the WYL conjugate gradient (CG) formula, with β WYL k ≥ 0, is further studied. A three-term WYL CG algorithm is presented, which has the sufficiently descent property without any conditions.…”

Section: When X Kmentioning

confidence: 99%

Unconstrained Optimization Methods: Conjugate Gradient Methods and Trust-Region Methods

Djordjević¹

2019

Applied Mathematics

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Here, we consider two important classes of unconstrained optimization methods: conjugate gradient methods and trust region methods. These two classes of methods are very interesting; it seems that they are never out of date. First, we consider conjugate gradient methods. We also illustrate the practical behavior of some conjugate gradient methods. Then, we study trust region methods. Considering these two classes of methods, we analyze some recent results.

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A Three-Term Conjugate Gradient Algorithm with Quadratic Convergence for Unconstrained Optimization Problems

Cited by 7 publications

References 37 publications

A Modified Dai–Liao Conjugate Gradient Method Based on a Scalar Matrix Approximation of Hessian and Its Application

A Modified Dai–Liao Conjugate Gradient Method Based on a Scalar Matrix Approximation of Hessian and Its Application

A New Paired Spectral Gradient Method to Improve Unconstrained and Non-Linear Optimization

Unconstrained Optimization Methods: Conjugate Gradient Methods and Trust-Region Methods

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