Abstract:
The paper is devoted to subgradient methods with switching between productive and nonproductive steps for problems of minimization of quasiconvex functions under functional inequality constraints. For the problem of minimizing a convex function with quasiconvex inequality constraints, a result is obtained on the convergence of the subgradient method with an adaptive stopping rule. Further, based on an analog of a sharp minimum for nonlinear problems with inequality constraints, results are obtained on the geometric convergence of restarted versions of subgradient methods. Such results are considered separately in the case of a convex objective function and quasiconvex inequality constraints, as well as in the case of a quasiconvex objective function and convex inequality constraints. The convexity may allow to additionally suggest adaptive stopping rules for auxiliary methods, which guarantee that an acceptable solution quality is achieved. The results of computational experiments are presented, showing the advantages of using such stopping rules.
The research of F.S. Stonyakin in Sections 2 and 5 was supported by the Strategic Academic Leadership Program "Priority 2030'"(agreement no. 075-02-2021-1316 of September 30, 2021). The research of A.V. Gasnikov in Section 4 was carried out under a state task of the Ministry of Science and Higher Education of the Russian Federation (project no. 0714-2020-0005).
Citation:
S. S. Ablaev, F. S. Stonyakin, M. S. Alkousa, A. V. Gasnikov, “Adaptive Subgradient Methods for Mathematical Programming Problems with Quasiconvex Functions”, Trudy Inst. Mat. i Mekh. UrO RAN, 29, no. 3, 2023, 7–25; Proc. Steklov Inst. Math. (Suppl.), 323, suppl. 1 (2023), S1–S18
This publication is cited in the following 2 articles:
S. M. Puchinin, E. R. Korolkov, F. S. Stonyakin, M. S. Alkusa, A. A. Vyguzov, “Cubgradientnye metody s shagom tipa B. T. Polyaka dlya zadach minimizatsii kvazivypuklykh funktsii s ogranicheniyami-neravenstvami i analogami ostrogo minimuma”, Kompyuternye issledovaniya i modelirovanie, 16:1 (2024), 105–122
S. S. Ablaev, A. N. Beznosikov, A. V. Gasnikov, D. M. Dvinskikh, A. V. Lobanov, S. M. Puchinin, F. S. Stonyakin, “On Some Works of Boris Teodorovich Polyak on the Convergence of Gradient Methods and Their Development”, Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, 64:4 (2024), 587