Slice convergence : stabilité et optimisation dans les espaces non réflexifs

Hajioui, Khalid El; Mentagui, Driss

ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 4, pp. 505-525.

Il est démontré par Mentagui [ESAIM : COCV 9 (2003) 297-315] que, dans le cas des espaces de Banach généraux, la convergence d'Attouch-Wets est stable par une classe d'opérations classiques de l'analyse convexe, lorsque les limites des suites d'ensembles et de fonctions satisfont certaines conditions de qualification naturelles. Ceci tombe en défaut avec la slice convergence. Dans cet article, nous établissons des conditions de qualification uniformes assurant la stabilité de la slice convergence et de la slice convergence duale par les mêmes opérations, dont le rôle est fondamental en optimisation convexe. Nous obtenons comme conséquences certains résultats clés de stabilité de l'épi-convergence établis par Mc Linden et Bergstrom [Trans. Amer. Math. Soc. 286 (1981) 127-142] en dimension finie. Comme application, nous présentons un modèle de convergence et de stabilité recouvrant une large classe de problèmes en optimisation convexe et en théorie de la dualité. Les éléments clés dans notre démarche sont l'analyse d'horizon, les notions de quasi-continuité et d'inf-locale compacité des fonctions convexes, puis la bicontinuité de la transformation de Legendre-Fenchel relativement à la slice convergence et la slice convergence duale.

It is shown by Mentagui [ESAIM: COCV 9 (2003) 297-315] that, in the case of general Banach spaces, the Attouch-Wets convergence is stable by a class of classical operations of convex analysis, when the limits satisfy some natural qualification conditions. This fails with the slice convergence. We establish here uniform qualification conditions ensuring the stability of the slice convergence under the same operations which play a basic role in convex optimization. We obtain as consequences, some key stability results of epi-convergence established by Mc Linden and Bergstrom [Trans. Amer. Math. Soc. 286 (1981) 127-142] in finite dimension. As an application, we give a model of convergence and stability for a wide class of problems in convex optimization and duality theory. The key ingredients in our methodology are, the horizon analysis, the notions of quasi-continuity and inf-local compactness of convex functions, and the bicontinuity of the Legendre-Fenchel transform relatively to the slice convergence.

[1] H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series. Pitman, London (1984). | MR | Zbl

[2] H. Attouch, D. Azé and R.J.-B. Wets, Convergence of convex-concave saddle functions; continuity properties of the Legendre-Fenchel transform with applications to convex programming and mechanics. Ann. Inst. Henri Poincaré 5 (1988) 537-572. | Numdam | MR | Zbl

[3] H. Attouch and G. Beer, On the convergence of subdifferentials of convex functions. Arch. Math. 60 (1993) 389-400. | MR | Zbl

[4] H. Attouch and H. Brezis, Duality for the sum of convex functions in general Banach spaces. Publications AVAMAC, Perpignan, 84-10. Av (1984).

[5] H. Attouch and R.J.-B. Wets, Quantitative stability of variational systems: II. A framework for nonlinear conditionning. IIASA working paper (1988) 88-89.

[6] H. Attouch and R. J. -B. Wets, Quantitative stability of variational systems: I. The epigraphical distance. Trans. Amer. Math. Soc. 328 (1991) 695-729. | MR | Zbl

[7] D. Azé and J.-P. Penot, Operations on convergent families of sets and functions. Optimization 21 (1990) 521-534. | MR | Zbl

[8] B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer, Nonlinear parametric optimization. Akademie Verlag (1982). | MR | Zbl

[9] G. Beer, Topologies on closed and closed convex sets and the Effros measurability of set valued functions, in Sém. d'Anal. Convexe, Montpellier (1991), exposé No. 2, 2.1-2.44. | Zbl

[10] G. Beer, The slice topology: A viable alternative to Mosco convergence in nonreflexive spaces. Nonlinear. Anal. Theo. Meth. Appl. 19 (1992) 271-290. | MR | Zbl

[11] G. Beer and J. Borwein, Mosco convergence and reflexivity. Proc. Amer. Math. Soc. 109 (1990) 427-436. | MR | Zbl

[12] G. Beer and R. Lucchetti, Convex optimization and the epi-distance topology. Trans. Amer. Math. Soc. 327 (1991) 795-813. | MR | Zbl

[13] G. Beer and R. Lucchetti, The epi-distance topology: Continuity and stability results with applications to convex optimization problems. Math. Oper. Res. 17 (1992) 715-726. | MR | Zbl

[14] G. Beer and R. Lucchetti, Weak topologies for the closed subsets of a metrizable space. Trans. Amer. Math. Soc. 335 (1993) 805-822. | MR | Zbl

[15] N. Bourbaki, Espaces vectoriels topologiques. Masson, Paris (1981). | MR | Zbl

[16] H. Brezis, Analyse fonctionnelle, théorie et applications. Masson, Paris (1983). | MR | Zbl

[17] C. Castaing and M. Valadier, Convex analysis and measurable multifunctions. Lect. Notes Math. 580 (1977). | MR | Zbl

[18] J. Dieudonné, Sur la séparation des ensembles convexes. Math. Annal. 163 (1966) 1-3. | EuDML | MR | Zbl

[19] S. Dolecki, G. Salinetti and R.J.-B. Wets, Convergence of functions: equi-semicontinuity. Trans. Amer. Math. Soc. 276 (1983) 409-429. | MR | Zbl

[20] A.L. Dontchev and T. Zolezzi, Well-posed optimization problems. Lect. Notes Math. 1543 (1993). | MR | Zbl

[21] I. Ekeland et R. Temam, Analyse convexe et problèmes variationnels. Dunod, Paris (1974). | MR | Zbl

[22] K. El Hajioui, Convergences variationnelles: approximations inf-convolutives généralisées, stabilité et optimisation dans les espaces non réflexifs. Thèse de Doctorat, Université Ibn Tofail, Kénitra (2002).

[23] K. El Hajioui et D. Mentagui, Sur la stabilité d'une convergence variationnelle dans les espaces de Banach généraux, en préparation.

[24] J. Hadamard, Sur les problèmes aux dérivées partielles et leur signification physique. Publ. Univ. Princeton 13 (1902) 49-52.

[25] J. Hadamard, Lectures on Cauchy's problem in linear partial differential equations. Dover (1953). | MR | Zbl

[26] J.L. Joly, Une famille de topologies et de convergences sur l'ensemble des fonctionnelles convexes. Thèse Grenoble (1970).

[27] G. Köthe, Topological vector spaces (I, II). Springer (1969, 1979). | MR | Zbl

[28] J. Lahrache, Stabilité et convergences dans les espaces non réflexifs, in Sém. d'Anal. Convexe Montpellier, exposé No. 10 (1991). | MR | Zbl

[29] P.J. Laurent, Approximation et optimisation. Hermann, Paris (1972). | MR | Zbl

[30] L. Mclinden and R.C. Bergstrom, Preservation of convergence of convex sets and functions in finite dimensions. Trans. Amer. Math. Soc. 286 (1981) 127-142. | MR | Zbl

[31] D. Mentagui, Inf-convolution polaire, stabilité de l'épi-convergence et estimation de la rapidité de convergence d'une suite de compacts. Thèse Rabat (1988).

[32] D. Mentagui, Problèmes d'optimisation biens posés et convergences variationnelles. Théorie et applications dans le cadre de l'optimisation non différentiable. Thèse d'État, F.U.N.D.P., Namur (1996).

[33] D. Mentagui, Caractérisation de la stabilité d'un problème de minimisation associé à une fonction de perturbation particulière. Pub. Inst. Math. 60 (1996) 65-74. | EuDML | MR | Zbl

[34] D. Mentagui, Analyse de récession et résultats de stabilité d'une convergence variationnelle. Application à la théorie de la dualité en programmation mathématique. ESAIM: COCV 9 (2003) 297-315. | EuDML | Numdam | MR | Zbl

[35] D. Mentagui et K. El Hajioui, Convergences des fonctions convexes et approximations inf-convolutives généralisées. Publ. Inst. Math., Nouvelle série 86 (2002) 123-136. | EuDML | MR | Zbl

[36] J.J. Moreau, Fonctionnelles convexes. Sém. sur les E.D.P. collège de France, Paris (1967). | Numdam

[37] U. Mosco, Approximation of the solutions of some variational inequalities. Ann. Scuola Normale Sup. Pisa 21 (1967) 373-394. | EuDML | Numdam | MR | Zbl

[38] U. Mosco, On the continuity of the Young-Fenchel transform. J. Math. Anal. Appl. 25 (1971) 518-535. | MR | Zbl

[39] R. Phelps, Convex functions, monotone operators and differentiability. Lect. Notes Math. 1364 (1989). | MR | Zbl

[40] H. Radström, An imbedding theorem for spaces of convex sets. Proc. Amer. Math. Soc. 3 (1952) 165-169. | MR | Zbl

[41] R.T. Rockafellar, Convex Analysis. Princeton Univ. Press (1970). | MR | Zbl

[42] R.T. Rockafellar and R.J.-B. Wets, Variational analysis. Springer (1998). | MR | Zbl

[43] Y. Sonntag and C. Zalinescu, Set convergences: An attempt of classification, in Proc. of Intl. Conf. on Diff. Equations and Control theory, Iasi, Romania, August (1990) 199-226. | MR | Zbl

[44] A.N. Tikhonov, Stability of inverse problems. Dokl. Akad. Nauk. USSR 39 (1943) 176-179. | MR | Zbl

[45] A.N. Tikhonov, Solution of incorrectly formulated problems and the regularization methods. Soviet Math. Dokl. 4 (1963) 1035-1038. | Zbl

[46] A.N. Tikhonov, Methods for the regularization of optimal control problems. Soviet Math. Dokl. 6 (1965) 761-763. | MR | Zbl

[47] A.N. Tikhonov and V. Arsenine, Methods for solving ill-posed problems. Nauka (1986). | MR

[48] R.J.-B. Wets, A formula for the level sets of epi-limits and some applications, Mathematical theories of optimization, J.P. Cecconi and T. Zolezzi Eds., Lect. Notes Math. 983 (1983). | MR | Zbl

[49] R.A. Wijsman, Convergence of sequences of convex sets, cones and functions. Bull. Amer. Math. Soc. 70 (1964) 186-188. | MR | Zbl

[50] R.A. Wijsman, Convergence of sequences of convex sets, cones and functions II. Trans. Amer. Math. Soc. 123 (1966) 32-45. | MR | Zbl

[51] T. Zolezzi, On stability in mathematical programming. Math. Programming 21 (1984) 227-242. | MR

[52] T. Zolezzi, Continuity of generalized gradients and multipliers under perturbations. Math. Oper. Res. 10 (1985) 664-673. | MR | Zbl

[53] T. Zolezzi, Stability analysis in optimization. Lect. Notes Math. 1990 (1986) 397-419. | MR | Zbl