## Towards minimal assumptions for the infimal convolution regularization.(English)Zbl 0759.49003

Summary: We encompass the Moreau-Yosida regularization process by infimal convolution into a general framework. This sheds light on the assumptions required for obtaining the usual properties. In particular the class of lower-$$T^ 2$$ mappings is shown to be a suitable class for performing the usual proximal regularization in open subsets of Hilbert spaces. The role of growth conditions is pointed out.

### MSC:

 49J45 Methods involving semicontinuity and convergence; relaxation

### Keywords:

infimal convolution; growth conditions
Full Text:

### References:

 [1] Attouch, H., Variational Convergence for Functions and Operators, (Applicable Math. Series (1984), Pitman: Pitman London) · Zbl 0561.49012 [2] Attouch, H.; Wets, R. J.-B, A convergence theory for saddle functions, Trans. Amer. Math. Soc., 280, No. 1, 1-41 (1983) · Zbl 0525.49009 [3] Bertsekas, D. P., Convexification procedures and decomposition methods for nonconvex optimization problems, J. Optim. Theory Appl., 29, 169-197 (1979) · Zbl 0389.90080 [4] Bougeard, M. L., Contribution à la Théorie de Morse en dimension finie, (Thèse de 3ème cycle (1978), Université Paris-Dauphine: Université Paris-Dauphine Paris) [5] Bougeard, M. L., Morse theory for some lower-$$C^2$$ functions in finite dimensions, Math. Programming, 41, No. 2, 141-169 (1988) · Zbl 0646.49029 [6] Castaing, C., A propos de l’existence des sections séparément mesurables et séparement semi-continues d’une multiapplication séparément semi-continue inférieurement, (Travaux du Séminaire d’Analyse Convexe. Travaux du Séminaire d’Analyse Convexe, Montpellier (1976)), 6.1-6.6, Exposé 6 · Zbl 0356.46045 [7] Clarke, F. H., Optimization and Nonsmooth Analysis (1983), Wiley: Wiley New York · Zbl 0727.90045 [8] De Giorgi, E., Convergence problems for functionals and operators, (De Giorgi, E.; Magenes, E.; Mosco, U., Proceedings, Int. Meeting on Recent Methods in Nonlinear Analysis. Proceedings, Int. Meeting on Recent Methods in Nonlinear Analysis, Roma, 1978 (1979), Pitagora: Pitagora Bologna), 131-188 [9] Dolecki, S., On inf-convolutions, generalized convexity and seminormality (1985), Université de Limoges, preprint [10] Ekeland, I., On the variational principle, J. Math. Anal. Appl., 47, 324-353 (1974) · Zbl 0286.49015 [11] Ekeland, I.; Lasry, J. M., On the number of closed trajectories for a Hamiltonian flow on a convex energy surface, Ann. of Math, 112, 283-319 (1980) · Zbl 0449.70014 [12] Fitzpatrick, S., Metric projections and the differentiability of distance functions, Bull. Austral. Math. Soc., 22, No. 2, 291-312 (1980) · Zbl 0437.46012 [13] Fougeres, A.; Truffert, A., Régularisation s.c.i. et Γ-convergence: Approximations inf-convolutives associées à un référentiel (version augmentée), Publications AVAMAC, Université de Perpignan 84-08/15 (1984) [14] Hiriart-Urruty, J.-B, Extension of Lipschitz functions, J. Math. Anal. Appl., 77, No. 2, 539-554 (1980) · Zbl 0455.26006 [15] Lasry, J.-M; Lions, P. L., Remark on regularization in Hilbert spaces, Israël J. Math., 55, No. 3, 257-266 (1986) · Zbl 0631.49018 [16] Nomizu, K.; Ozeki, H., The existence of complete Riemannian metrics, (Proc. Sympos. Pure Math. Soc., 12 (1961)), 889-891 · Zbl 0102.16401 [17] Palais, R. S., Critical point theory and the minimax principle, (Chern, S.; Smale, S., Global Analysis. Global Analysis, Proc. Symp. Pure Math., Vol. 15 (1970), Amer. Math. Soc.,: Amer. Math. Soc., Providence, RI), 185-212 · Zbl 0212.28902 [18] Penot, J.-P, Calcul sous-différentiel et optimisation, J. Funct. Anal., 27, 248-276 (1978) · Zbl 0404.90078 [19] Penot, J.-P, A characterization of tangential regularity, Nonlinear Anal., 5, No. 6, 625-643 (1981) · Zbl 0472.58010 [20] Penot, J.-P, On favorable classes of mappings in nonliner analysis and optimization (1985), Univ. Pau, preprint [21] Penot, J.-P; Bougeard, M.-L, Approximation and decomposition properties of some classes of locally d.c. functions, Math. Programming, 41, No. 2, 195-228 (1988) · Zbl 0666.49005 [22] Pommellet, A., Analyse convexe et théorie de Morse, (Thèse de 3ème cycle (1982), Université Paris IX) [23] Rockafellar, R.-T, Integral functionals, normal integrands and measurable selections, (Waelbroeck, L., Nonlinear Operators and the Calculus of Variations. Nonlinear Operators and the Calculus of Variations, Lecture Notes in Math., Vol. 543 (1976), Springer-Verlag: Springer-Verlag Berlin/New York), 157-207 · Zbl 0374.49001 [24] Rockafellar, R.-T, Favorable classes of Lipschitz continuous functions in subgradient optimization, (Nurminski, E., Progress in Nondifferentiable Optimization (1982), I.I.A.S.A.,: I.I.A.S.A., Laxemburg, Australia), 125-143 · Zbl 0511.26009 [25] Rockafellar, R.-T; Wets, R. J.-B, Variational systems, an introduction, (Salinetti, G., Multifunctions and Integrands. Multifunctions and Integrands, Lecture Notes in Mathematics, Vol. 1091 (1984), Springer-Verlag: Springer-Verlag Berlin/New York), 1-54 [26] Volle, M., Conjugaisons par tranches, Ann. Mat. Pura Appl., 139, 279-312 (1985) [28] Wets, R., Convergence of convex functions, variational inequalities and convex optimization problems, (Cottle, P.; etal., Variational Inequalities and Complementary Problems (1980), Wiley: Wiley Chichester) · Zbl 0481.90066
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.