Two adaptive RMIL-TYPE conjugate gradient parameters derived by minimizing ℓ₁ and ℓ∞ norm condition numbers of its search direction matrix

Abstract

This study introduces two adaptive scaling parameters for the Rivaie–Mustafa–Ismail–Leong (RMIL) type conjugate gradient framework designed for unconstrained optimization problems. The proposed parameters are obtained by minimizing the condition numbers of the RMIL-type search direction matrix under the and matrix norms, respectively. This minimization strategy is intended to enhance the conditioning and numerical stability of the search direction, thereby improving the overall performance of the conjugate gradient algorithm. The resulting adaptive parameters, denoted by  and , are expressed in explicit form using gradient differences  and search directions . To assess performance, numerical experiments are conducted on 57 CUTEr benchmark problems with dimensions varying from 50 to 100,000. The results are evaluated using performance profiles based on iteration counts, total function and gradient evaluations, and CPU time. The computational outcomes indicate that the parameter leads to a marked improvement in the efficiency of the RMIL-type method, whereas the parameter yields comparatively weaker performance. Overall, the findings highlight the effectiveness of adaptive parameter selection driven by matrix norm conditioning in conjugate gradient methods.