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Subgradient selector in the generalized cutting plane method with an application to sparse optimization
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DOI：10.1007/s11590-025-02246-w.png)
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Abstract
En 中文
Duality in convex analysis devotes a prominent role to affine functions, as proper convex lower semicontinuous functions are supremum of such functions. This property is used in the Kelley's algorithm, to minimize a proper convex lower semicontinuous function by sequentially approximating it from below by maxima of affine functions (cuts). Affine functions are deduced from a bilinear pairing. In generalized convexity, the usual bilinear form is replaced by some bivariate function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c$$\end{document}, called coupling. The Moreau-Rockafellar subdifferential of a function is replaced by the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c$$\end{document}-subdifferential. Kelley's algorithm then becomes the generalized \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c$$\end{document}-cutting plane method to minimize a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c$$\end{document}-subdifferentiable objective function. In this paper, we prove a convergence result whose scope makes it possible to tackle sparse optimization problems. For this purpose, we introduce a selection of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c$$\end{document}-subgradients involved in a pointwise locally equicontinuous property, together with the coupling \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c$$\end{document} and the objective function. Under the assumptions of the convergence result, we discuss a necessary condition on the continuity points of the function to be minimized. Finally, we give an example of converging Capra-cutting plane method for the minimization of the pseudonorm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _0$$\end{document} on a compact set.
Keywords:
Abstract convexity
Capra
Cutting plane
Generalized convexity
Sparsity
Journal
O
IF:
1.1
Papers:
72
Citations:
2.4K