A Modified Three-Term Hestenes–Stiefel Conjugate Gradient Method with Guaranteed Descent Property

Hybridization of two conjugate gradient methods for multiobjective optimization beyond convexity and Lipschitz continuity

A three-term conjugate gradient method with a new descent structure for unconstrained optimization and low-carbon supply chain optimization

Optimization methods play a vital role in solving large-scale problems across engineering and environmental fields. Among them, the conjugate gradient (CG) method is popular for its computational efficiency, and recent developments—such as the three-term CG variant—have shown improved convergence and numerical stability. This paper proposes an improved three-term CG method for unconstrained optimization. Inspired by the Three-Term Zheng-Huang-Shi (ZHS) CG parameterization and Three-Term Rivaie-Mustafa-Ismail-Leong (TTRMIL), this method is specifically designed to improve the computational performance of CG methods. In the proposed method, sufficient descent conditions and global convergence properties for general functions are mathematically proven under assumption and criteria from the strong Wolfe line search. Numerical experiments conducted on several unconstrained optimization problems highlight the superiority of the new method over certain CG methods with similar characteristics. In real-world applications, the proposed method is extended to address the challenge of reducing carbon emissions for sustainability, particularly within the Low Carbon Supply Chain (LCSC) optimization problem.

Global convergence of a coderivative-based regularized Newton method with damping for nonsmooth optimization

A globally convergent regularized Newton method with positive definite regularization for solving nonsmooth optimization problems that replaces the identity matrix in traditional algorithms with a general positive-definite symmetric matrix to regularize the generalized Hessian.

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

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.

A Wolfe-Type Spectral Conjugate Gradient Method for Nonsmooth Convex Optimization Problems

This paper proposes a Wolfe-type spectral conjugate gradient method for nonsmooth convex optimization, built on the Moreau-Yosida regularization of the objective function. The method combines a safeguarded spectral parameter with a Dai-Kou-type conjugate parameter, and uses a Wolfe-type line search compatible with the inexact gradients that the regularization produces. We establish global convergence of the method, together with an R-linear convergence rate under an additional strong-convexity assumption. The method is evaluated on standard nonsmooth optimization benchmarks and on large-scale problems, and compared against several existing conjugate gradient and bundle-type methods. The results show that the proposed method performs competitively overall, matching or outperforming existing methods on most problems tested, while a few specific limitations of the current implementation are also identified and discussed

A Novel Fractional-Order Accelerated Gradient Descent Method for Nonlinear Optimization with Application to Posture Recognition

This article proposes a Caputo fractional accelerated gradient descent (CFAGD) method for unconstrained optimization problems that is applicable to both smooth and a class of non-smooth objective functions. The proposed approach incorporates an adaptive (\b{eta}k)-parameter, which is heuristically updated throughout the iterative process to improve the search direction. Furthermore, the method employs the Caputo fractional derivative together with the adaptive (\b{eta}^k)-parameter, thereby preserving the memory characteristics associated with non-integer-order derivatives. A suitable step-size is selected via an inexact line-search technique based on the Armijo condition. The central idea is to scale the step-size by a positive parameter to improve the behavior of the iterates as they approach an optimal point, thereby generating a descent sequence. Under strong convexity and bounded Hessian assumptions, linear convergence of the proposed method is established. Numerical validations, including neural-network-based examples, further indicate that the CFAGD method can achieve faster and more stable performance than competing approaches.