A New Subspace Minimization Conjugate Gradient Algorithm Without Explicit Use of Gradient Terms

Motivated by the Nazareth conjugate gradient method and the subspace minimization technique, we propose a new subspace minimization conjugate gradient algorithm that eliminates explicit gradient terms from the subspace basis. The algorithm constructs a subspace using the previous search direction and the gradient differences across three consecutive iterates. Within this subspace, a quadratic approximation problem is solved to determine the subsequent search direction, and the global convergence result is established under standard Wolfe line search conditions. Numerical experiments demonstrate that, compared to existing methods such as CG\_DESCENT\cite{CGDESCENT} and GGOPT\cite{CGOPT}, the proposed algorithm solves a higher percentage of problems with fewer iterations, reduced CPU runtime, and fewer function and gradient evaluations, highlighting its robustness and efficiency.