Subgradient Selector in the Generalized Cutting Plane Method with an Application to Sparse Optimization

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

Abstract: 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 algorithm, to minimize a proper convex lower semicontinuous function by sequentially approximating it from below by maxima of affinne 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 subdierential of a function is replaced by the $c$-subdierential. The Kelley algorithm then becomes the generalized c-cutting plane method to minimize a $c$-subdierentiable 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.