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A new class of memory gradient methods with inexact line searches. (English) Zbl 1072.65086

The aim of this study is to present a new class of memory gradient methods with inexact line searches for unconstrained minimization problems. In order to accelerate the convergence rate of algorithms and to avoid additional function evaluations, the proposed techniques use more previous iterative information than other methods to generate a search direction and perform inexact line searches to select an appropriate step-size at each iteration. The new method does need storing and computing matrices (associated with the Hessian of objective functions) at each iteration, as in ‘memory-less’ quasi-Newton methods, thus it proves to be suitable especially to solve large scale unconstrained minimization problems.
The author shows that, under weak mild conditions, the new class of memory gradient methods has the desired property of global convergence. The convergence rate is investigated in the case of special objective functions, and the method is proved to converge more stably than similar approaches, being available to solve some ill-conditioned problems. A series of comparative numerical experiments illustrates the effectiveness of the proposed memory gradient methods in solving unconstrained minimization problems. Using principally the Armijo’s rule and Wolfe’s rule as inexact line search rules, the computational results are favourably compared to other classical methods such as Fletcher-Reeves, Polak-Ribière, Hestenes-Stiefel conjugate gradient methods and steepest descent method. The new algorithm is shown to behave effectively stable, and to provide robust and practical computational qualities when solving large scale minimization problems.

MSC:

65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
90C52 Methods of reduced gradient type
90C53 Methods of quasi-Newton type
90C06 Large-scale problems in mathematical programming

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References:

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