### TWO NOVEL GRADIENT METHODS WITH OPTIMAL STEP SIZES

Harry Oviedo, Oscar Dalmau, Rafael Herrera

1. Centro de Investigación en Matemáticas, CIMAT A. C. Guanajuato, Gto. Mexico
• Received:2018-09-29 Revised:2019-02-18 Published:2021-04-12
• Contact: Oscar Dalmau,Email:dalmau@cimat.mx
• Supported by:
This work was supported in part by CONACYT (Mexico), Grants 258033 and 256126.

Harry Oviedo, Oscar Dalmau, Rafael Herrera. TWO NOVEL GRADIENT METHODS WITH OPTIMAL STEP SIZES[J]. Journal of Computational Mathematics, 2021, 39(3): 375-391.

In this work we introduce two new Barzilai and Borwein-like steps sizes for the classical gradient method for strictly convex quadratic optimization problems. The proposed step sizes employ second-order information in order to obtain faster gradient-type methods. Both step sizes are derived from two unconstrained optimization models that involve approximate information of the Hessian of the objective function. A convergence analysis of the proposed algorithm is provided. Some numerical experiments are performed in order to compare the efficiency and effectiveness of the proposed methods with similar methods in the literature. Experimentally, it is observed that our proposals accelerate the gradient method at nearly no extra computational cost, which makes our proposal a good alternative to solve large-scale problems.

CLC Number:

 [1] El-Sayed M.E. Mostafa. A CONJUGATE GRADIENT METHOD FOR DISCRETE-TIME OUTPUT FEEDBACK CONTROL DESIGN [J]. Journal of Computational Mathematics, 2012, 30(3): 279-297. [2] N.H.Sweilam,L.F.Abd-Elal. A COMPUTATIONAL APPROACH FOR OPTIMAL CONTROL SYSTEMS GOVERNED BY PARABOLIC VARIATIONAL INEQUALITIES$^*$ [J]. Journal of Computational Mathematics, 2003, 21(6): 815-824. [3] Yu Hong DAI,Qin NI. TESTING DIFFERENT CONJUGATE GRADIENT METHODS FOR LARGE-SCALE UNCONSTRAINED OPTIMIZATION [J]. Journal of Computational Mathematics, 2003, 21(3): 311-320.