Variational metric and exponential penalization. (English) Zbl 0688.90043

H. Attouch and R. J. Wets [Trans. Am. Math. Soc. 296, 33-60 (1986; Zbl 0607.49009)] have introduced recently a variational metric between closed proper convex functions. The aim of this note is to give an estimation of this metric in the case of exponential penalties. We can therefore recover some convergence results for the exponential penalty method.
Reviewer: K.Mouallif


90C25 Convex programming
65K05 Numerical mathematical programming methods
49M30 Other numerical methods in calculus of variations (MSC2010)


Zbl 0607.49009
Full Text: DOI


[1] Murphy, F.,A Class of Exponential Penalty Functions, SIAM Journal on Control, Vol. 12, pp. 679-687, 1974. · Zbl 0289.90038
[2] Attouch, H., andWets, R.,Isometries for the Legendre-Fenchel Transform, Transactions of the American Mathematical Society, Vol. 296, pp. 33-60, 1986. · Zbl 0607.49009
[3] Brezis, H.,Opérateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, Holland, 1973.
[4] Attouch, H.,Variational Convergence for Functions and Operators, Pitman, New York, New York, 1984. · Zbl 0561.49012
[5] Aubin, J. P.,L’Analyse Nonlinéaire et ses Motivations Economiques, Masson, Paris, France, 1984. · Zbl 0551.90001
[6] Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970. · Zbl 0193.18401
[7] Rockafellar, R. T.,Monotone Operators and the Proximal Point Algorithm, SIAM Journal on Control and Optimization, Vol. 14, pp. 877-896, 1976. · Zbl 0358.90053
[8] Ekeland, I., andTemam, R.,Analyse Convexe et Problèmes Variationnels, Dunod-Gauthier-Villars, Paris, France, 1974.
[9] Lemaire, B.,Coupling Optimization Methods and Variational Convergence: Trends in Mathematical Optimization, Edited by K. H. Hoffmann, J. B. Hiriart-Urruty, C. Lemarechal, and J. Zowe, Birkhaüser Verlag, Basel, Switzerland, Vol. 84, pp. 163-179, 1988.
[10] Mouallif, K., andTossings, P., Une Méthode de Pénalisation Exponentielle Associée à une Régularisation Proximale, Bulletin de la Société Royale des Sciences de Liège, Vol. 56, pp. 181-192, 1987.
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