Derivative-Free Hybrid Conjugate Gradient Method for Constrained Nonlinear Equations with Projection in Line Searches

Abstract

 Robustness, efficiency, and accuracy are qualities that excellent algorithms should have. Due to the simplicity and minimal storage requirements, Conjugate gradient (CG) methods are useful for solving large-scale, unconstrained optimization problems. Despite that, it has a few drawbacks. Even if they have high numerical performance, certain approaches lack global convergence properties; therefore, the solutions might not be the most accurate. Various methods and modifications have been done. Some formulations would be difficult to comprehend and apply, and would lead to high CPU time. The proving process would also be impacted by the complex formulations. Over the past years, researchers have developed various globally convergent CG methods, but with a complicated algorithm, it rather hampered the implementation. Therefore, new CG methods with a derivate-free approach that have good convergence properties and outperform the existing CG coefficients in terms of number of iterations (NOI), number of function evaluations (NFE), and central processing time per unit (CPU time) are proposed. The proposed method will employ a non-derivative approach. This approach should make the algorithm’s processing time as minimal as possible. The comparison for derivative-free tools among the existed derivative-free CG. The proposed approach was chosen because it combines the strengths of the CG method with derivate-free optimization to optimize complicated objective functions without explicitly computing derivatives. This paper will show the derivate-free CG, which was proven to fulfil both convergence analysis and numerical performance.