Gradient Projection and Conditional Gradient Methods for Constrained Nonconvex Minimization

Minimization of a smooth function on a sphere or, more generally, on a smooth manifold, is the simplest non-convex optimization problem. It has a lot of applications. Our goal is to propose a version of the gradient projection algorithm for its solution and to obtain results that guarantee convergen...

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Bibliographic Details
Published in:Numerical functional analysis and optimization Vol. 41; no. 7; pp. 822 - 849
Main Authors: Balashov, M. V., Polyak, B. T., Tremba, A. A.
Format: Journal Article
Language:English
Published: Abingdon Taylor & Francis 18.05.2020
Taylor & Francis Ltd
Subjects:
90C52. Secondary: 46N10
Algorithms
Computational geometry
Convergence
Convexity
Frank-Wolfe method
gradient projection method
Ležanski-Polyak-Lojasiewicz condition
Manifolds
Minimization on a sphere
nonconvex optimization
Nonlinear programming
Optimization
proximally smooth set
smooth functions
strongly convex set
ISSN:0163-0563, 1532-2467
Online Access:Get full text
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Summary:Minimization of a smooth function on a sphere or, more generally, on a smooth manifold, is the simplest non-convex optimization problem. It has a lot of applications. Our goal is to propose a version of the gradient projection algorithm for its solution and to obtain results that guarantee convergence of the algorithm under some minimal natural assumptions. We use the Ležanski-Polyak-Lojasiewicz condition on a manifold to prove the global linear convergence of the algorithm. Another method well fitted for the problem is the conditional gradient (Frank-Wolfe) algorithm. We examine some conditions which guarantee global convergence of full-step version of the method with linear rate.
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ISSN:0163-0563
1532-2467
DOI:10.1080/01630563.2019.1704780