A Modified Three-Term Hestenes–Stiefel Conjugate Gradient Method with Guaranteed Descent Property
Conjugate gradient (CG) methods are among the most widely used iterative algorithms for large-scale unconstrained optimization, valued for their simplicity and minimal memory requirements. A central challenge is ensuring guaranteed sufficient descent of the search direction, independent of the line search procedure. This paper proposes a Modified Three-Term Hestenes–Stiefel (MTTHS) conjugate gradient method that enforces the sufficient descent condition exactly, without any line search restriction. The proposed direction incorporates a geometric correction term θₖyₖ into the standard three-term framework, where the scalar θₖ is derived analytically to satisfy gₖ₊₁ᵀdₖ₊₁ = −‖gₖ₊₁‖². Global convergence is established under the strong Wolfe line search conditions via the Zoutendijk convergence criterion. Numerical experiments were based on 23 test problems, each solved at five different dimensions, n ∈ {100, 500, 1000, 5000, 10000}, yielding 115 test cases. The results show that the MTTHS method is robust and computationally efficient