A Modified Three-Term Hestenes–Stiefel Conjugate Gradient Method with Guaranteed Descent Property
TL;DR
A Modified Three-Term Hestenes–Stiefel (MTTHS) conjugate gradient method that enforces the sufficient descent condition exactly, without any line search restriction is proposed.
Abstract
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