×

Approximation and regularization of arbitrary functions in Hilbert spaces by the Lasry-Lions method. (English) Zbl 0780.41021

Summary: The Lasry-Lions regularization method is extended to arbitrary functions. In particular, to any proper lower semicontinuous functions \(f: X\to \mathbb{R} \cup\{+\infty\}\) defined on a Hilbert space \(X\) and which is quadratically minorized (i.e. \(f(x)\geq -c(1+ \| x\|^ 2)\) for some \(c\geq 0\)), is associated a sequence of differentiable functions with Lipschitz continuous derivatives which approximate \(f\) from below. Some variants of the method are considered including the case of non quadratic kernels.

MSC:

41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
65K10 Numerical optimization and variational techniques
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] Asplund, E., Fréchet differentiability of convex functions, Acta Math., Vol. 121, 31-47 (1968) · Zbl 0162.17501
[2] Attouch, H., Variational Convergences for Functions and Operators (1984), Applicable Mathematics Series: Applicable Mathematics Series Pitman London · Zbl 0561.49012
[3] Attouch, H.; Azé, D.; Wets, R. J.B., On continuity properties of the partial Legendre-Fenchel transform: convergence of sequences of augmented Lagrangians functions, Moreau-Yosida approximates and subdifferential operators, (Hiriart-Urruty, J.-B., Fermat days 85: Mathematics for Optimization (1986), North-Holland: North-Holland Amsterdam), 1-42
[4] Attouch, H.; Wets, R. J.B., Epigraphical analysis, (Attouch, H.; Aubin, J.-P.; Clarke, F. H.; Ekeland, I., Analyse non linéaire (1989), Gauthier-Villars: Gauthier-Villars Paris), 73-100 · Zbl 0676.49003
[6] Bougeard, M., Contribution à la théorie de Morse en dimension finie, Thèse de
[7] Penot, J.-P.; Bougeard, M., Approximation and decomposition properties of some classes of locally d.c. functions, Math. Progr, Vol. 41, 195-227 (1989) · Zbl 0666.49005
[8] Bougeard, M.; Penot, J.-P.; Pommelet, A., Towards minimal assumptions for the infimal convolution regularization, J. of Approx. Theory, Vol. 64, 245-270 (1991) · Zbl 0759.49003
[9] Brézis, H., Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert (1973), North-Holland: North-Holland Amsterdam
[10] Clarke, F. H., Optimization and Nonsmooth Analysis (1983), J. Wiley: J. Wiley New York · Zbl 0727.90045
[11] Ekeland, I.; Lasry, J.-M., Problèmes variationnels non convexes en dualité, C. R. Acad. Sci. Paris, 291, Series I, 493-497 (1980) · Zbl 0448.90063
[12] Hiriart-Urruty, J.-B., A general formula on the conjugate of the difference of functions, Can. Math. Bull., Vol. 29, 482-485 (1986) · Zbl 0608.90087
[13] Hiriart-Urruty, J.-B.; Plazanet, Ph, Moreau’s decomposition Theorem revisited, (Attouch, H.; Aubin, J.-P.; Clarke, F. H.; Ekeland, I., Analyse non linéaire (1989), Gauthier-Villars: Gauthier-Villars Paris) · Zbl 0675.90093
[14] Hiriart-Urruty, J.-B., Extension of Lipschitz functions, f. Math. Anal. Appi., Vol. 72, 539-554 (1980) · Zbl 0455.26006
[15] Hiriart-Urruty, J.-B., Lipschitz r-continuity of the approximate subdifferential of a convex function, Math. Scand., Vol. 47, 123-134 (1980) · Zbl 0426.26005
[16] Lasry, J.-M.; Lions, P.-L., A remark on regularization in Hilbert spaces, Israel Journal of Mathematics, Vol. 55, 257-266 (1986) · Zbl 0631.49018
[17] Moreau, J.-J., Fonctionnelles convexes, Lecture Notes (1967), Collège de France: Collège de France Paris
[18] Moreau, J.-J., Proximité et dualité dans un espace Hilbertien, Bull. Soc. Math. Fr., Vol. 93, 273-299 (1965) · Zbl 0136.12101
[19] Pazy, A., Semi-groups of nonlinear contractions in Hilbert spaces, Problems in Nonlinear Analysis, 343-430 (1971), Edizioni Cremonese: Edizioni Cremonese Roma · Zbl 0228.47038
[20] Pommelet, A., Analyse convexe et théorie de Morse, Thèse de
[21] Rockafellar, R. T., Convex Analysis (1966), Princeton University Press · Zbl 0145.15901
[22] Rockafellar, R. T., Generalized directional derivatives and subgradients of nonconvex functions, Canadian J. Math., Vol. 32, 157-180 (1980) · Zbl 0447.49009
[24] Vladimirov, A. A.; Nesterov, Yu. E.; Chekanov, Yu. N., On uniformly convex functionals, Vest. Mosk. Univ., Vol. 3, 12-23 (1978), (Russian) · Zbl 0442.47046
[25] Wells, J. C., Difierentiable functions on Banach spaces with lipschitz derivative, J. Differential Geometry, Vol. 8, 135-152 (1973) · Zbl 0289.58005
[26] Yosida, K., Functional analysis (1971), Springer, Berlin, Heidelberg: Springer, Berlin, Heidelberg New York · Zbl 0217.16001
[27] Zalinescu, C., On uniformly convex functions, Jour. Math. Anal. Appi., Vol. 95, 344-374 (1983) · Zbl 0519.49010
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.