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JOURNALS // Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki // Archive

Zh. Vychisl. Mat. Mat. Fiz., 2024 Volume 64, Number 4, Pages 676–692 (Mi zvmmf11738)

Papers published in the English version of the journal

The gradient projection method for a supporting function on the unit sphere and its applications

M. V. Balashova, A. A. Trembaab

a V.A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, 117997, Moscow, Russia
b Moscow Institute of Physics and Technology (National Research University), 141701, Dolgoprudny, Moscow oblast, Russia

Abstract: We consider minimization of the supporting function of a convex compact set on the unit sphere. In essence, this is the problem of projecting zero onto a compact convex set. We consider sufficient conditions for solving this problem with a linear rate using a first order algorithm—the gradient projection method with a fixed step-size and with Armijo’s step-size. We consider some applications for problems with set-valued mappings. The mappings in the work basically are given through the set-valued integral of a set-valued mapping with convex and compact images or as the Minkowski sum of finite number of convex compact sets, e.g., ellipsoids. Unlike another solution ways, e.g., with approximation in a certain sense of the mapping, the considered algorithm much weaker depends on the dimension of the space and other parameters of the problem. It also allows efficient error estimation. Numerical experiments confirm the effectiveness of the considered approach.

Key words: gradient projection method, set-valued integral, sum of ellipsoids, strong convexity, uniform convexity, nonsmooth analysis.

Received: 22.05.2023
Revised: 07.11.2023
Accepted: 07.06.2024

Language: English


 English version:
Computational Mathematics and Mathematical Physics, 2024, 64:4, 676–692


© Steklov Math. Inst. of RAS, 2025