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Armijo's rule, Barzilai-Borwein method, Conjugate gradient methods, Nonmonotone linesearch, Unconstrianed optimization.
Abstract
The gradient method is the basis for many nonlinear optimization methods, and is also one of the methods used to solve the large scale unconstrained optimization. It requires a small storage volume compared to its counterparts. In this paper we have given a detailed presentation of the gradient method with Armijo's rule, and then a methods to improve its performance. This is represented in the Barzilai-Borwein method, which provides us with the step length the long of the steepest descent direction without the need for linesearch, but, the Barzilai-Borwein method is not always convergent. To solve this problem, we presented an algorithm attributed to the researcher Marcos Raydan, which linked the Barzilai-Borwein method with the nonmonotone linesearch of Grippo-Lampariello-Lucidi. In the end, we conducted numerical tests of the previous methods through a Matlab computer program.
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