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As we know, the stepsize is extremely crucial to gradient method. A new type of stepsize is introduced for the gradient method in the paper, which is generated by minimizing the norm of the approximate model of the gradient along the line of negative gradient direction. Based on the retard technique, we present a new gradient method by minimizing adaptively the approximate model of the objective function and the norm of the approximate model of the gradient along the line of negative gradient direction for strictly convex quadratic minimization. The convergence of the proposed method is established. The numerical experiments on four groups of strictly convex quadratic minimization problems illustrate that the proposed method is very promising. We also extend the new gradient method for convex quadratic minimization to general unconstrained optimization by incorporating a nonmonotone line search. The convergence of the resulting method is established. The numerical experiment on the 147 test functions from the CUTEst library indicates that the resulting method is superior to some efficient gradient methods including the BBQ method (SIAM J Optim. 2021;31(4):3068–3096) and is competitive to two famous conjugate gradient software packages CGOPT (1.0) and CG $ \_ $ _ DESCENT (5.0). [ABSTRACT FROM AUTHOR]

A new subspace minimization conjugate gradient method based on tensor model is proposed and analysed. If the objective function is close to a quadratic, we construct a quadratic approximation model in a two-dimensional subspace to generate the search direction; otherwise, we construct a tensor model. It is remarkable that the search direction satisfies the sufficient descent property. We prove the global convergence of the proposed method under mild assumptions. Numerical comparisons are given with well-known CGOPT and CG_DESCENT and show that the proposed algorithm is very promising. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new concept of approximate optimal stepsize for gradient method, use it to interpret the Barzilai-Borwein (BB) method, and present an efficient gradient method with approximate optimal stepsize for large unconstrained optimization. If the objective function f is not close to a quadratic on a line segment between the current iterate xk and the latest iterate xk−1, we construct a conic model to generate the approximate optimal stepsize for gradient method if the conic model is suitable to be used. Otherwise, we construct a new quadratic model or two other new approximation models to generate the approximate optimal stepsize for gradient method. We analyze the convergence of the proposed method under some suitable conditions. Numerical results show the proposed method is very promising. [ABSTRACT FROM AUTHOR]

In this paper, we present a new conjugate gradient method, in which the search direction is computed by minimizing a selected approximate model in a two-dimensional subspace. That is, if the objective function is not close to a quadratic, the search direction is generated by a conic model. Otherwise, a quadratic model is considered. The direction of the proposed method is proved to possess the sufficient descent property. With the modified nonmonotone line search, we establish a global convergence of the proposed method under appropriate assumptions. R-linear convergence of the proposed method is also analyzed. Numerical results using two different test function collections show that the proposed algorithm is efficient. [ABSTRACT FROM AUTHOR]