A New Subspace Minimization Conjugate Gradient Method for Unconstrained Minimization

Abstract

Subspace minimization conjugate gradient (SMCG) methods are a class of quite efficient iterative methods for unconstrained optimization and have received increasing attention recently. The search directions of SMCG methods are generated by minimizing an approximate model with the approximate matrix $B_k$ over the two-dimensional subspace spanned by the current gradient $g_k$ and the latest step. The main drawback of SMCG methods is that the parameter $g_k^TB_kg_k$ in the search directions must be determined when calculating the search directions. The parameter $g_k^TB_kg_k$ is crucial to SMCG methods and is difficult to be determined properly. An alternative solution for this drawback might be to exploit a new way to derive SMCG methods independent of $g_k^TB_kg_k$. The projection technique has been used successfully to derive conjugate gradient directions such as the Dai–Kou conjugate gradient direction (Dai and Kou in SIAM J Optim 23(1):296–320, 2013). Motivated by the above two observations, we use a projection technique to derive a new SMCG method independent of $g_k^TB_kg_k$. More specifically, we project the search direction of memoryless quasi-Newton method into the above two-dimensional subspace and derive a new search direction, which is proved to be descent. Remarkably, the proposed method without any line search enjoys the finite termination property for two-dimensional strictly convex quadratic functions. An adaptive scaling factor in the search direction is exploited based on the finite termination property. The proposed method does not need to determine the parameter $g_k^TB_kg_k$ and can be regarded as an extension of the Dai–Kou conjugate gradient method. The global convergence of the proposed method is established under the suitable assumptions. Numerical comparisons on the 147 test functions from the CUTEst library indicate that the proposed method is very promising.