arXiv subjects

Ravi Raushan

Publications and source records attributed to Ravi Raushan.

3 recordsLinked to original sources

arXiv2026

Proximal Gradient Methods for Unconstrained Set Optimization Problems with Set-Valued Maps of Finite Cardinality

This work presents two different types of proximal gradient methods, with line search and without line search, for solving unconstrained set-valued optimization problems under the lower set-less ordering relation induced by a solid cone that is convex, pointed, and closed. The objective mapping of the problem involves finitely many functions, with each one being the sum of a continuously differentiable function and a convex function that is proper and closed. We present an approach to characterize weakly minimal points of the problem with the help of weakly efficient points of a family of vector optimization problems. Thereafter, we establish a stationarity condition along with its connection with weakly minimal points of the problem under study. Based on the stationary condition, the concept of a descent direction at a non-stationary point is discussed. In view of the line search-based method, we formulate an Armijo-type line search condition and establish the existence of such a step-size. For the proposed methods, global convergence is established under mild assumptions. The convergence analysis of the proximal gradient method with line search provides a theoretical advancement over the convergence results previously established for the steepest descent method in set-valued optimization problems. In addition, we analyze the computational complexity of the proposed methods and show that both methods achieve a convergence rate of $\mathcal{O}(1/\sqrt{k})$. Numerical results are reported to test the performance of the methods in practice.

math.OC↗

arXiv2026

Hager-Zhang Conjugate Gradient Method for Set Optimization with Set-Valued Objective Map of Finite Cardinality

This work introduces a nonlinear Hager-Zhang conjugate gradient method for solving set optimization problems. The objective function under consideration is defined by a finite collection of continuously differentiable functions. Notably, the proposed approach imposes restrictions neither on the existence of a finite generator of the ordering cone nor on any regularity condition at the optimal solution. As a result, the proposed method holds considerable significance for both set optimization and vector optimization problems, with the latter serving as a special case of the former. The study begins by discussing Wolfe line search conditions using Drummond-Svaiter scalarization function. Thereafter, we establish the existence of a step length satisfying the Wolfe line search conditions along a descent direction. The Hager-Zhang scalar conjugate parameter is introduced to derive the search direction for the proposed method. It is established that the direction generated by the proposed method is a descent direction. The well-definedness of the proposed method is given. Furthermore, we discuss some important results and a Zoutendijk-like condition to ensure global convergence. Subsequently, the global convergence of the proposed method is established in an asymptotic manner. Finally, numerical experiments on various test problems validate the practical performance and effectiveness of the proposed technique.

math.OC↗

arXiv2024

Nonlinear Conjugate Gradient Methods for Optimization of Set-Valued Mappings of Finite Cardinality

This article presents nonlinear conjugate gradient methods for finding local weakly minimal points of set-valued optimization problems under a lower set less ordering relation. The set-valued objective function of the optimization problem under consideration is defined by finitely many continuously differentiable vector-valued functions. For such optimization problems, at first, we propose a general scheme for nonlinear conjugate gradient methods and then introduce Dai-Yuan, Polak-Ribi{\`e}re-Polyak, and Hestenes-Stiefel conjugate gradient parameters for set-valued functions. Toward deriving the general scheme, we introduce a condition of sufficient decrease and Wolfe line searches for set-valued functions. For a given sequence of descent directions of a set-valued function, it is found that if the proposed standard Wolfe line search technique is employed, then the generated sequence of iterates for set optimization follows a Zoutendijk-like condition. With the help of the derived Zoutendijk-like condition, we report that all the proposed nonlinear conjugate gradient schemes are globally convergent under usual assumptions. It is important to note that the ordering cone used in the entire study is not restricted to be finitely generated, and no regularity assumption on the solution set of the problem is required for any of the reported convergence analyses. Finally, we demonstrate the performance of the proposed methods through numerical experiments. In the numerical experiments, we demonstrate the effectiveness of the proposed methods not only on the commonly used test instances for set optimization but also on a few newly introduced problems under general ordering cones that are neither nonnegative hyper-octant nor finitely generated.

math.OC↗