Optimizing Non-Linear Fuzzy Functions Using LS and RMIL Methods

The Liu–Storey (LS) nonlinear conjugate gradient method, although structurally similar to the well-known Polak–Ribiere–Polyak (PRP) and Hestenes–Stiefel (HS) methods, has not been extensively explored in the literature. This paper introduces a novel LS-type method inspired by the memoryless BFGS quasi-Newton approach, offering guaranteed global convergence for general functions when combined with the Grippo–Lucidi line search. Furthermore, we propose a modified LS method that ensures global convergence for nonconvex minimization problems when utilizing the strong Wolfe line search. Complementary numerical results validate these theoretical claims. In parallel, this paper also presents the RMIL (Robust Maximum Influence Linearity) method, a new approach for nonlinear analysis using advanced optimization techniques. The RMIL method addresses the challenges of modeling complex, nonlinear relationships by maximizing influence linearity within a robust optimization framework, significantly improving model accuracy and stability. This dual-method approach leverages both LS-type and RMIL methodologies, demonstrating enhanced predictive performance and robustness in nonlinear systems across various applications, including specific field. The integration of these advanced methods highlights their potential for solving complex optimization problems in nonlinear analysis.