TY - GEN A1 - Vu, Thi Huong A1 - Koch, Thorsten A1 - Xu, Hong-Kun T1 - The gradient projection method: Is the Polyak adaptive stepsize rule optimal? N2 - Not always! This is our answer to the question of whether the Polyak adaptive stepsize rule in the gradient projection method is optimal. The answer is based on revisiting the subgradient projection method by Polyak [USSR Computational Mathematics and Mathematical Physics 9 (1969)] for smooth and convex minimization problems where the objective function possesses a geometric property called flatness. Our results show that the method can be more flexible (the effective range for the parameter controlling the stepsize can be wider) and have sharper convergence rates. Applications to split feasibility/equality problems are presented, deriving for the first time the O(1/k) rate of convergence for the adaptive CQ method. A theoretical guarantee of the linear convergence of the gradient descent method with adaptive stepsizes for Google PageRank is provided. At the same time, numerical experiments are designed to spot the ``optimal" stepsize and to compare with other basic gradient methods. Y1 - 2024 ER - TY - JOUR A1 - Vu, Thi Huong A1 - Xu, Hong-Kun A1 - Nguyen, Dong Yen T1 - Stability analysis of split equality and split feasibility problems JF - Journal of Global Optimization N2 - In this paper, for the first time in the literature, we study the stability of solutions of two classes of feasibility (i.e., split equality and split feasibility) problems by set-valued and variational analysis techniques. Our idea is to equivalently reformulate the feasibility problems as parametric generalized equations to which set-valued and variational analysis techniques apply. Sufficient conditions, as well as necessary conditions, for the Lipschitz-likeness of the involved solution maps are proved by exploiting special structures of the problems and by using an advanced result of B.S. Mordukhovich [J. Global Optim. 28, 347–362 (2004)]. These conditions stand on a solid interaction among all the input data by means of their dual counterparts, which are transposes of matrices and regular/limiting normal cones to sets. Several examples are presented to illustrate how the obtained results work in practice and also show that the assumption on the existence of a nonzero solution used in the necessity conditions cannot be lifted. Y1 - 2025 U6 - https://doi.org/10.1007/s10898-025-01469-6 SN - 0925-5001 VL - 92 SP - 411 EP - 429 PB - Springer Science and Business Media LLC ER - TY - JOUR A1 - Vu, Thi Huong A1 - Xu, Hong-Kun A1 - Nguyen, Dong Yen T1 - Stability of nonhomogeneous split equality and split feasibility problems with possibly nonconvex constraint sets JF - Optimization N2 - By applying some techniques of set-valued and variational analysis, we study solution stability of nonhomogeneous split equality problems and nonhomogeneous split feasibility problems, where the constraint sets need not be convex. Necessary and sufficient conditions for the Lipschitz-likeness of the solution maps of the problems are given and illustrated by concrete examples. The obtained results complement those given in [Huong VT, Xu HK, Yen ND. Stability analysis of split equality and split feasibility problems. arXiv:2410.16856.], where classical split equality problems and split feasibility problems have been considered. Y1 - 2025 U6 - https://doi.org/10.1080/02331934.2025.2508499 SP - 1 EP - 19 ER -