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It is widely accepted that the stepsize is of great significance to gradientmethod. Two efficient gradient methods with approximately optimal stepsizesmainly based on regularization models are proposed for unconstrainedoptimization. More exactly, if the objective function is not close to aquadratic function on the line segment between the current and latest iterates,regularization models are exploited carefully to generate approximately optimalstepsizes. Otherwise, quadratic approximation models are used. In particular,when the curvature is non-positive, special regularization models aredeveloped. The convergence of the proposed methods is established under theweak conditions. Extensive numerical experiments indicated the proposed methodis superior to the BBQ method (SIAM J. Optim. 2021,31(4), 3068-3096) and otherefficient gradient methods, and is competitive to two famous and efficientconjugate gradient software packages CG$ \_ $DESCENT (5.0) (SIAM J. Optim.16(1), 170-192, 2005) and CGOPT (1.0) (SIAM J. Optim. 23(1), 296-320, 2013) forthe CUTEr library. Due to the surprising efficiency, we believe that gradientmethods with approximately optimal stepsizes can become strong candidates forlarge-scale unconstrained optimization.

Two efficient gradient methods with approximately optimal stepsizes based on regularization models for unconstrained optimization