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In this paper, four accelerated subspace minimization conjugate gradient methods based on 2-regularization or 3-regularization models with different matrices are proposed. The search directions are generated by minimizing a quadratic approximation and different regularization models of the objective function, and the new directions satisfy the sufficient descent condition. In addition, an acceleration technique is used to improve the stepsize. With a modified nonmonotone line search, we establish the global convergence of the proposed methods under mild assumptions. By using the Kurdyka-Łojasiewicz property, we analyze the convergence rates of function sequence and iterative sequence without the convexity of objective function. Numerical experiments show that the proposed methods are very effective, especially for CUTEr test problems with no less than 50 dimensions, the best of the new methods is competitive to the latest limited memory conjugate gradient software package CG_DESCENT(6.8) developed by Hager and Zhang (SIAM J. Optim. 23:2150–2168, 24). [ABSTRACT FROM AUTHOR]

In this paper, two new subspace minimization conjugate gradient methods based on p-regularization models are proposed, where a special scaled norm in p-regularization model is analyzed. Different choices of special scaled norm lead to different solutions to the p-regularized subproblem. Based on the analyses of the solutions in a two-dimensional subspace, we derive new directions satisfying the sufficient descent condition. With a modified nonmonotone line search, we establish the global convergence of the proposed methods under mild assumptions. R-linear convergence of the proposed methods is also analyzed. Numerical results show that, for the CUTEr library, the proposed methods are superior to four conjugate gradient methods, which were proposed by Hager and Zhang (SIAM J. Optim. 16(1):170–192, 2005), Dai and Kou (SIAM J. Optim. 23(1):296–320, 2013), Liu and Liu (J. Optim. Theory. Appl. 180(3):879–906, 2019) and Li et al. (Comput. Appl. Math. 38(1):2019), respectively. [ABSTRACT FROM AUTHOR]