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Abstract In this paper, we first propose a new three-term conjugate gradient (CG) method, which is based on the least-squares technique, to determine the CG parameter, named LSTT. And then, we present two improved variants of the LSTT CG method, aiming to obtain the global convergence property for general nonlinear functions. The least-squares technique used here well combines the advantages of two existing efficient CG methods. The search directions produced by the proposed three methods are sufficient descent directions independent of any line search procedure. Moreover, with the Wolfe–Powell line search, LSTT is proved to be globally convergent for uniformly convex functions, and the two improved variants are globally convergent for general nonlinear functions. Preliminary numerical results are reported to illustrate that our methods are efficient and have advantages over two famous three-term CG methods.

In this paper, we investigated a new improved Conjugate Gradient (CG) algorithm of a Three-Term type (TTCG) based on Dai and Liao procedure to improve the CG algorithm of (Hamoda, Rivaie, and Mamat / HRM). The new CG-algorithm satisfies both the conjugacy condition and the sufficient descent condition. The step-size of this TTCG-algorithm would be computed by accelerating the Wolfe-Powell line search technique. The proposed new TTCG algorithms have demonstrated their global affinity in certain specific circumstances given in this paper.

In this paper, we first propose a new three-term conjugate gradient (CG) method, which is based on the least-squares technique, to determine the CG parameter, named LSTT. And then, we present two improved variants of the LSTT CG method, aiming to obtain the global convergence property for general nonlinear functions. The least-squares technique used here well combines the advantages of two existing efficient CG methods. The search directions produced by the proposed three methods are sufficient descent directions independent of any line search procedure. Moreover, with the Wolfe–Powell line search, LSTT is proved to be globally convergent for uniformly convex functions, and the two improved variants are globally convergent for general nonlinear functions. Preliminary numerical results are reported to illustrate that our methods are efficient and have advantages over two famous three-term CG methods. [ABSTRACT FROM AUTHOR]

Abstract The conjugate gradient (CG) method is one of the most popular methods to solve nonlinear unconstrained optimization problems. The Hestenes-Stiefel (HS) CG formula is considered one of the most efficient methods developed in this century. In addition, the HS coefficient is related to the conjugacy condition regardless of the line search method used. However, the HS parameter may not satisfy the global convergence properties of the CG method with the Wolfe-Powell line search if the descent condition is not satisfied. In this paper, we use the original HS CG formula with a mild condition to construct a CG method with restart using the negative gradient. The convergence and descent properties with the strong Wolfe-Powell (SWP) and weak Wolfe-Powell (WWP) line searches are established. Using this condition, we guarantee that the HS formula is non-negative, its value is restricted, and the number of restarts is not too high. Numerical computations with the SWP line search and some standard optimization problems demonstrate the robustness and efficiency of the new version of the CG parameter in comparison with the latest and classical CG formulas. An example is used to describe the benefit of using different initial points to obtain different solutions for multimodal optimization functions.

The conjugate gradient (CG) method is one of the most popular methods to solve nonlinear unconstrained optimization problems. The Hestenes-Stiefel (HS) CG formula is considered one of the most efficient methods developed in this century. In addition, the HS coefficient is related to the conjugacy condition regardless of the line search method used. However, the HS parameter may not satisfy the global convergence properties of the CG method with the Wolfe-Powell line search if the descent condition is not satisfied. In this paper, we use the original HS CG formula with a mild condition to construct a CG method with restart using the negative gradient. The convergence and descent properties with the strong Wolfe-Powell (SWP) and weak Wolfe-Powell (WWP) line searches are established. Using this condition, we guarantee that the HS formula is non-negative, its value is restricted, and the number of restarts is not too high. Numerical computations with the SWP line search and some standard optimization problems demonstrate the robustness and efficiency of the new version of the CG parameter in comparison with the latest and classical CG formulas. An example is used to describe the benefit of using different initial points to obtain different solutions for multimodal optimization functions. [ABSTRACT FROM AUTHOR]