  Subspace minimization conjugate gradient (SMCG) methods have become a class
of quite efficient iterative methods for unconstrained optimization and have
attracted extensive attention recently. Usually, the search directions of SMCG
methods are generated by minimizing approximate models with the approximation
matrix $ B_k $ of the objective function at the current iterate over the
subspace spanned by the current gradient $ g_k $ and the latest search
direction. The $ g_k^TB_kg_k $ must be estimated properly in the calculation of
the search directions, which is crucial to the theoretical properties and the
numerical performance of SMCG methods. It is a great challenge to estimate it
properly. The projection technique has been used successfully to generate
conjugate gradient directions such as Dai-Kou conjugate gradient direction.
Motivated by the above two observations, in the paper we present a new subspace
minimization conjugate gradient methods by using a projection technique based
on the memoryless quasi-Newton method. More specially, we project the search
direction of the memoryless quasi-Newton method into the subspace spanned by
the current gradient and the latest search direction and drive 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 convex quadratic functions, which is helpful for designing
algorithm. An adaptive scaling factor in the search direction is given based on
the above finite termination property. The proposed method does not need to
determine the parameter $ \rho_k $ and can be regarded as an extension of
Dai-Kou conjugate gradient method. The global convergence of the proposed
method is analyzed. Numerical comparisons indicate the proposed method is very
promising.
