Pré-Publication, Document De Travail Année : 2024

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

Résumé

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 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 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 ℓ 0 on a compact set.

Keywords cutting plane method; generalized convexity; c-subdierential; pseudonorm ℓ 0 ; subgradient selector 1 * CERMICS, École nationale des ponts et chaussées, IP Paris, France 1 spirit of so-called cutting plane methods, which sequentially minimize and update maxima of base functions (generalized cuts).

Elementary base functions are the building block of so-called abstract convexity where the equivalent of closed convex functions are the suprema of such functions. By contrast, in generalized convexity, the focus is put on replacing the usual bilinear form (of duality in convex analysis) by some bivariate function c, that is called coupling, and the Moreau-Rockafellar subdierential of a function by the c-subdierential. Abstract and generalized convexity are two (related) ways to extend duality beyond convex analysis. As such, they provide the mathematical framework to extend cutting plane methods beyond the convex case.

The cutting plane method is a staple optimization scheme in integer linear programming [14, 5]. In 1958, Gomory [6] introduced one of the first cutting plane methods to solve integer linear programs. In 1960, Kelley [7] proposed a cutting plane method to minimize convex functions (not necessarily dierentiable) over a compact set. Pallaschke and Rolewicz [11, Theorem 9.1.1] generalized Kelley's result to the minimization of so-called Φconvex functions, where elementary base functions are continuous functions. Rubinov gave two convergence results [13, Propositions 9.2, 9.3] in the abstract convex setting. Each of these convergence results relies on properties which relate elementary base functions (generalized cuts), the objective function and all generalized subgradients. In the usual convex nite dimensional setting and also for so-called one-sided-linear couplings, as introduced in [2, 2.2] all of these assumptions boil down to boundedness of the generalized subdierential of the objective function. Now, it has been established in [2] that the ℓ 0 pseudonorm which counts the number of nonzero entries of a vector has nonempty generalized subdierential, for a suitable choice of elementary base functions, induced by a so-called Euclidean Capra-coupling. Unfortunately, the assumptions of results [11, 13] are not straightforwardly satised by the Capra coupling and the ℓ 0 pseudonorm, as the Capra-subdierentials of ℓ 0 have the property of being unbounded. This observation has motivated us to extend the scope of the results in [11, 13]. In particular, we emphasize the choice of suitable subsets of subgradients in the statement of a generalized cutting plane method. With this, it is possible to tackle sparse optimization problems, consisting in minimizing ℓ 0 over a compact set. More precisely, as easy-to-compute formulas for the Capra-subdierential of ℓ 0 are given in [8], we can design a converging Capra-cutting plane method with the corresponding Capra-cuts.

Contributions. We introduce the notion of dual selector and, in Theorem 2, we propose a convergence result of the generalized cutting plane method for c-subdierentiable functions that satisfy a pointwise locally equicontinuous property. We discuss the link between this property and the continuity points of the objective function in Proposition 3. Finally, we present a converging Capra-cutting plane method for the minimization of ℓ 0 over a compact subset of the unit sphere in Proposition 8.

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hal-04856783 , version 1 (27-12-2024)

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  • HAL Id : hal-04856783 , version 1

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Seta Rakotomandimby, Jean-Philippe Chancelier, Michel de Lara, Adrien Le Franc. Subgradient Selector in the Generalized Cutting Plane Method with an Application to Sparse Optimization. 2024. ⟨hal-04856783⟩

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