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<p>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&#xa0;<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(c\)</EquationSource> </InlineEquation>, called coupling. The Moreau-Rockafellar subdifferential of a function is replaced by the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(c\)</EquationSource> </InlineEquation>-subdifferential. Kelley’s algorithm then becomes the generalized <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(c\)</EquationSource> </InlineEquation>-cutting plane method to minimize a <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(c\)</EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(c\)</EquationSource> </InlineEquation>-subgradients involved in a pointwise locally equicontinuous property, together with the coupling&#xa0;<InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(c\)</EquationSource> </InlineEquation> 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&#xa0;<InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\ell _0\)</EquationSource> </InlineEquation> on a compact set.</p>

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Subgradient selector in the generalized cutting plane method with an application to sparse optimization

  • 期刊论文

  • 发表日期：

    2025年11月8日

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  • Seta Rakotomandimby,
  • Jean-Philippe Chancelier,
  • Michel De Lara,
  • Adrien Le Franc

摘要

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  \(c\) , called coupling. The Moreau-Rockafellar subdifferential of a function is replaced by the \(c\) -subdifferential. Kelley’s algorithm then becomes the generalized \(c\) -cutting plane method to minimize a \(c\) -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 \(c\) -subgradients involved in a pointwise locally equicontinuous property, together with the coupling  \(c\) 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  \(\ell _0\) on a compact set.