Approximation and decomposition properties of some classes of locally d.c. functions.(English)Zbl 0666.49005

We study the connexion between local and global decompositions of some important subclasses of locally d.c. functions (functions which locally split as a difference of two convex functions). Then we tackle the problem of regularizing such functions by the Moreau-Yosida process and prove in particular that the class of lower-$$C^ 2$$ functions fits well this approximation procedure.

MSC:

 49J52 Nonsmooth analysis 26B25 Convexity of real functions of several variables, generalizations 90C30 Nonlinear programming 49M27 Decomposition methods 90C55 Methods of successive quadratic programming type
Full Text:

References:

 [1] E. Asplund, ”Fréchet differentiability of convex functions,”Acta Mathematica 121 (1968) 31–47. · Zbl 0162.17501 [2] H. Attouch,Variational Convergence for Functions and Operators, Applicable Mathematics Series (Pitman, London, 1984)). · Zbl 0561.49012 [3] H. Attouch, ”Variational properties of epiconvergence. Application to limit analysis problems in mechanics and duality theory,” in: G. Salinetti, ed.Multifunctions and Integrands, Lecture Notes in Mathematics 1091 (Springer-Verlag, Berlin, 1985). [4] J. Baranger, ”Existence de solutions pour des problèmes d’optimisation non convexes,”Journal de Mathématiques Pures et Appliquées 52 (1973) 377–406. · Zbl 0274.46015 [5] J. Baranger and R. Temam: ”Nonconvex optimization problems depending on a parameter,”SIAM Journal on Control 13 (1975) 146–152. · Zbl 0302.49003 [6] D.P. Bertsekas, ”Convexification procedures and decomposition methods for nonconvex optimization problems,”Journal of Optimization Theory and Applications 29 (1979) 169–197. · Zbl 0389.90080 [7] M. Bougeard, ”Contribution à la Théorie de Morse en dimension finie,” Thése troisième cycle, University Paris-IX-Dauphine (Paris, 1978). [8] M. Bougeard, ”Contribution à la Théorie de Morse,” Cahier Ceremade, University Paris-Dauphine (Paris, 1979). [9] M.L. Bougeard, ”About critical points of some lowerC 2 functions” in: C. Lemarechal, ed.,Third Franco-German Conference in Optimization (I.N.R.I.A., 78153 Le Chesnay, France, 1984) pp. 12–16. [10] H. Brezis,Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, Mathematics Studies 5 (North-Holland, Amsterdam, 1973). [11] C. Castaing and M. Valadier,Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580 (Springer-Verlag, Berlin, 1977). · Zbl 0346.46038 [12] F.H. Clarke,Optimization and Nonsmooth Analysis (Wiley, New York, 1983). · Zbl 0582.49001 [13] B. Cornet, Contribution à la théorie mathématique des mécanismes dynamiques d’allocation des ressources, Thèse d’Etat, University Paris IX (1981). [14] I. Ekeland and J.-M. Lasry, ”On the number of periodic trajectories for a hamiltonian flow,”Annals of Mathematics 112 (1980) 293–319. · Zbl 0449.70014 [15] I. Ekeland, and J.-M. Lasry, ”Problèmes variationnels non convexes en dualité,”Comptes rendus de l’Académie des Sciences de Paris A 291 (1980) 493–496. [16] I. Ekeland, and G. Lebourg, ”Generic Fréchet differentiability and pertubed optimization problems in Banach spaces,”Transactions American Mathematical Society 224 (4) (1976) 193–216. · Zbl 0313.46017 [17] I. Ekeland, and R. Temam,Analyse Convexe et Problèmes Variationnels (Dunod, Gauthier-Villars, Paris, 1972 (English translation: North-Holland, Amsterdam, American Elsevier, New York, 1976). · Zbl 0281.49001 [18] R. Ellaia, Contribution à l’analyse et l’optimisation de différence de fonctions convexes, Thèse de troisiéme cycle, University of Toulouse (1984). [19] A. Fougeres, and A. Truffert, ”Régularisation s.c.i. et {$$\Gamma$$}-convergence. Approximations inf-convolutives associées à un référentiel” (version augmentée), Publications AVAMAC, University of Perpignan 84–08/15 (1984). [20] P. Hartman, ”On functions representable as a difference of convex functions,”Pacific Journal of Mathematics 9 (1959) 707–713. · Zbl 0093.06401 [21] J.-B. Hiriart-Urruty, ”Extension of Lipschitz functions.”Journal of Mathematical Analysis and Applications 77 (2) (1980) 539–554. · Zbl 0455.26006 [22] J.-B. Hiriart-Urruty, ”Generalized differentiability, duality and optimization for problems dealing with differences of convex functions,” Written version of a lecture given in Groningen, Preprint University of Toulouse (1984). [23] H. Holmes, ”Smoothness of certain metric projections in Hilbert spaces,”Transactions of the American Mathematical Society 183 (1973) 87–100. · Zbl 0268.46043 [24] R. Janin, ”Sur la dualité et la sensibilité dans les problèmes de programmes mathématiques,” Thèse d’Etat, Paris VI (1974). [25] H. Th. Jongen, P. Jonker and F. Twilt,Nonlinear Optimization in $$\mathbb{R}$$n (book to appear). · Zbl 0985.90083 [26] W.A. Kirk, ”Caristi’s fixed point theorem and metric convexity,”Colloquium Mathematicum 36 (1976) 81–86. · Zbl 0353.53041 [27] B. Lacolle, ”Un procédé d’approximation d’une fonction convexe lipschitzienne et de ses singularités,” Preprint, University of Grenoble (1984). [28] J.-M. Lasry and P.-L. Lions,Remark on regularization in Hilbert spaces, Cahier Ceremade no. 8414, University Paris-Dauphine (1984). [29] C. Lescarret, ”Application ”prox” dans un espace de Banach,”Comptes rendus Académie des Sciences Paris 265 (1967) 676–678. · Zbl 0175.15203 [30] C. Malivert, ”Méthode de descente sur un fermé non convexe,”Bulletin Société Mathématique de France Mémoire 60 (1979) 113–124. · Zbl 0416.49020 [31] C. Malivert, J.-P. Penot and M. Thera, ”Minimisation d’une fonction régulière sur un fermé régulier non convexe,”Comptes Rendus de l’Académie des Sciences Paris A (1978) 1191–1193. · Zbl 0382.58010 [32] J.-J. Moreau, ”Proximité et dualité dans un espace hilbertien,”Bulletin de la Sociéte Mathématique de France 93 (1965) 273–299. [33] J.-J. Moreau, ”Fonctionnelles convexes,” Séminaire sur les Equations aux Dérivées partielles, Collége de France, Paris (1967). [34] P. Michel, ”Problèmes des inégalités; applications à la programmation et au contrôle optimal,”Bulletin Société Mathématique de France 101 (1973) 413–439. · Zbl 0295.90033 [35] A. Pazy, ”Semi-groups of nonlinear contractions in Hilbert spaces,” in:Problems in Non-linear Analysis, (Centro Italiano Matematico Estivo IV ciclo, Varenna, 1970; Cremonese, Rome, 1971), 343–430. [36] J.-P. Penot, ”Sous-différentiels de fonctions numériques non convexes,”Comptes rendus de l’Académie des Sciences Paris A 278 (1974) 1153–1155. [37] J.-P. Penot, ”Penalization and regularization methods in nonsmooth analysis,” Unpublished Lecture, Journées d’Optimisation de Louvain, Louvain (1983). [38] J.-P. Penot, ”Modified and augmented Lagrangian theory revisited and augmented,” Lecture in Journées Fermat de Toulouse, France, May 1985. [39] J.-P. Penot, ”On favorable classes of mappings in nonlinear analysis and optimization,” Preprint, University of Pau (1985). [40] A. Pommelet, ”Analyse convexe et théorie de Morse,” Thèse troisième cycle, University Paris IX (1982). [41] B.N. Pshenichnyii,Necessary Conditions for an Extremum (M. Dekker, New York, 1971). [42] A. Ralambo, ”Problèmes de minimisation dans les espaces de Banach,” Thèse troisième cycle, University Paris VI (June 1985). [43] R.T. Rockafellar, ”Augmented Lagrange multiplier functions and duality in nonconvex programming,”SIAM Journal on Control 12 (1974) 268–285. · Zbl 0285.90063 [44] R.T. Rockafellar: ”Favorable classes of Lipschitz continuous functions in subgradient optimization,” in: E. Nurminski, ed.,Progress in Nondifferentiable Optimization (IIASA, Laxenburg, Austria, 1982) pp. 125–143. · Zbl 0511.26009 [45] R.T. Rockafellar, R.J.-B. Wets, ”Variational systems, an introduction,” in: G. Salinetti, ed.,Multifunctions and Integrands, Lecture Notes in Mathematics 1091, (Springer-Verlag 1984) pp. 1–54. [46] J.E. Spingarn, ”Submonotone subdifferentials of Lipschitz functions,”Transactions of the American Mathematical Society 264 (1981) 77–89. · Zbl 0465.26008 [47] J.E. Spingarn, ”Submonotone mappings and the proximal algorithm,”Numerical Functional Analysis and Optimization 4 (2) (1981/1982) 123–150. · Zbl 0495.49025 [48] J.P. Vial, ”Strong and weak convexity of sets and functions,”Mathematics of Operations Research 8 (1983) 231–257. · Zbl 0526.90077 [49] R. Wets, ”Convergence of convex functions, variational inequalities and convex optimization problems,” in: Cottle et al., eds.,Variational Inequalities and Complementary Problems (Wiley, Chichester, 1980)). · Zbl 0481.90066 [50] D. Wexler, ”Prox-mappings associated with a pair of Legendre conjugate functions,”Revue francaise d’Automatique, Informatique, Recherche Opérationnelle R2 (1972) 39–65. · Zbl 0294.90075 [51] Y. Yomdin, ”On functions representable as a supremum of smooth functions,”S.I.A.M. Journal of Mathematical Analysis 14 (1983) 239–246. · Zbl 0524.26010 [52] Y. Yomdin, A. Shapiro, ”On functions representable as a difference of two convex functions and necessary conditions in constrained optimization,” preprint, University Neger Beer-Sheva (Israel, 1981).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.