Abstract
We provide calculus rules for global approximate minima concerning usual operations on functions. The formulas we obtain are then applied to approximate subdifferential calculus. In this way, new results are presented, for example on the approximate subdifferential of a deconvolution, or on the subdifferential of an upper envelope of convex functions.
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References
Abdulfattah, S. and Soueycatt, M. (1991),Analyse epi-/hypo-graphique, Séminaire d'Analyse convexe, Montpellier.
Attouch, H. (1990), Analyse épigraphique, Notes de cours de D.E.A., Montpellier.
Attouch, H. and Wets, R. (1989), Epigraphical Analysis,Analyse Nonlinéaire, eds. H. Attouch, J.-P. Aubin, F.H. Clarke, I. Ekeland, 73–100, Gauthier-Villars, Paris.
Benoist, J. and Hiriart-Urruty, J.-B. (1992), What is the Subdifferential of the Closed Convex Hull of a Function? Preprint.
Debreu, G. (1959), Theory of Value, John Wiley, New York.
Dedieu, J.-P. (1978), Critère de fermeture pour l'image d'un fermé non convexe par une multiapplication,C.R.A.S. Paris 287, 941–943.
Dubowicki, A. and Milyutin, A. (1965), Extremum Problems in the Presence of Restrictions,Comput. Math. and Math. Phys. 5, 1–80.
Goossens, P. (1984), Asymptotically Compact Sets, Asymptotic Cone and Closed Conical Hull,Bull. Soc. Roy. Se. Liège 1, 57–67.
Hiriart-Urruty, J.-B. (1982), ɛ-Subdifferential Calculus, inResearch Notes in Math.57, Pitman Publishers, New York, 43–92.
Hiriart-Urruty, J.-B. (1986), A General Formula on the Conjugate of the Difference of Functions,Canad. Math. Bull. 294, 482–485.
Hiriart-Urruty, J.-B. and Mazure, M.-L. (1986), Formulations variationnelles de l'addition parallèle et de la soustraction parralèle d'opérateurs semi-définis positifs,C.R.A.S. Paris 302, série I, nℴ 15, 527–530.
Hiriart-Urruty, J.-B. and Phelps, R.R. (1992), Applications of cɛ-Subdifferentials to the Calculus of Subdifferentials, Preprint.
Hiriart-Urruty, J.-B. and Lemarechal, C. (1993), Convex Analysis and Minimization Algorithms I, Springer-Verlag.
Kusraev, A. G. and Kutateladze, S. S. (1987),Subdifferential Calculus, Nauka Publishing House Novosibirsk (in Russian).
Laurent, P.-J. (1972),Approximation et optimisation, Hermann, Paris.
Martinez-Legaz, J.-E. and Seeger, A. (1992), A Formula on the Approximate Subdifferential of the Difference of Convex Functions,Bull. Austr. Math. Soc. 45, nℴ1, 37–41.
Mazure, M.-L. and Volle, M. (1991), Equations inf-convolutives et conjugaison de Moreau-Fenchel,Ann. Fac. Sci. de Toulouse XII, nℴ1, 103–126.
Moreau, J.-J. (1970), Inf-convolution, sous-additivité, convexité des fonctions numériques,J. Math, pures et appl. 49, 109–154.
Moussaoui, M. and Seeger, A. (1993), Sensitivity Analysis of Optimal Value Functions of Convex Parametric Programs with Possibly Empty Solution Sets, to appear inS.I.A.M. J. on Optimization.
Rockafellar, R.T. (1966), Extension of Fenchel's Duality Theorem for Convex Functions,Duke Math. J. 33, 81–90.
Rockafellar, R.T. (1970), Convex Analysis, Princeton University Press.
Rockafellar, R.T. (1974),Conjugate Duality and Optimization, SIAM Publications, Philadephia.
Singer, I. (1970), A Fenchel-Rockafellar Type Duality Theorem for Maximization,Bull. Aust. Math. Soc. 29, 193–198.
Toland, J. (1979), A Duality Principle for Nonconvex Optimization and the Calculus of Variations,Arch. Rational Mech. Anal. 71, 41–61.
Valadier, M. (1969), Sous-différentiels d'une borne supérieure et d'une somme continue de fonctions convexes,C.R.A.S. Paris 268, Série A, 39–42.
Volle, M. (1988), Concave Duality: Application to Problems Dealing with Difference of Functions,Math. Prog. 41, 261–278.
Volle, M. (1992), Some Applications of the Attouch-Brezis Condition to Closedness Criterions, Optimization and Duality, Séminaire d'Analyse convexe de Montpellier, 22, exposé nℴ16.
Volle, M. (1993) A Formula on the Subdifferential of the Deconvolution of Convex Functions,Bull. Austr. Math. Soc. 47(2), 333–340.
Volle, M. (1993), Sous-difffentiel d'une enveloppe supérieure de fonctions convexes,C.R.A.S. Paris, t. 317, Série I, 845–849.
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Volle, M. Calculus rules for global approximate minima and applications to approximate subdifferential calculus. J Glob Optim 5, 131–157 (1994). https://doi.org/10.1007/BF01100690
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DOI: https://doi.org/10.1007/BF01100690