Abstract
In this paper, we present a generalization of the Hessian matrix toC 1,1 functions, i.e., to functions whose gradient mapping is locally Lipschitz. This type of function arises quite naturally in nonlinear analysis and optimization. First the properties of the generalized Hessian matrix are investigated and then some calculus rules are given. In particular, a second-order Taylor expansion of aC 1,1 function is derived. This allows us to get second-order optimality conditions for nonlinearly constrained mathematical programming problems withC 1,1 data.
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References
Araya Schulz R, Gormaz Arancibia R (1979) Problemas localmente Lipschitzianos en optimizacion. (Master's thesis) University of Chile at Santiago
Auslender A (1979) Penalty methods for computing points that satisfy second order necessary conditions. Math Programming 17:229–238
Auslender A (1981) Stability in mathematical programming with nondifferentiable data; second order directional derivative for lower-C 2 functions.
Ben-Tal A, Zowe J (1982) Necessary and sufficient optimality conditions for a class of nonsmooth minimization problems. Math Programming 24:70–91
Brezis H (1973) Opérateurs maximaux monotones. North-Holland, Amsterdam
Bernstein B, Toupin R (1962) Some properties of the Hessian matrix of a strictly convex function. Journal für die Reine und Angewandte Mathematik 210:65–72
Clarke FH (1975) Generalized gradients and applications. Trans Amer Math Soc 205:247–262
Clarke FH (1976) On the inverse function theorem. Pacif J Math 64:97–102
Fitzpatrick S, Phelps R (to appear) Differentiability of the metric projection in Hilbert spaces.
Hestenes M (1975) Optimization theory: The finite dimensional case. Wiley, New York
Hiriart-Urruty J-B (1977) Contributions à la programmation mathématique: Cas déterministe et stochastique. (Doctoral thesis) Sciences Mathématiques, Université de Clermont-Ferrand II
Hiriart-Urruty J-B (1979) Refinements of necessary optimality conditions in nondifferentiable programming I. Appl Math Optim 5:63–82
Hiriart-Urruty J-B (1982a) Refinements of necessary optimality conditions in nondifferentiable programming II. Math Programming Study 19:120–139
Hiriart-Urruty J-B (1982b) Characterizations of the plenary hull of the generalized Jacobian matrix. Math Programming Study 17:1–12
Holmes R (1973) Smoothness of certain metric projections on Hilbert spaces. Trans Amer Math Soc 184:87–100
Ioffe A (1981) Second order conditions in nonlinear nonsmooth problems of semi-infinite programming.
Kinderlehrer D, Stampacchia G (1980) An introduction to variational inequalities and their applications. Academic Press, New York
Moreau J-J (1965) Proximité et dualité dans un espace hilbertien. Bulletin de la Société Mathématique de France 93:273–299
Ortega JM, Rheinboldt WC (1970) Iterative solution of nonlinear equations in several variables. Academic Press, New York
Rockafellar RT (1976) Solving a nonlinear programming problem by way of a dual problem. Symposia Mathematica 19:135–160
Stoer J, Witzgall C (1970) Convexity and optimization in finite dimensions I. Springer-Verlag, Berlin
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Communicated by J. Stoer
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Hiriart-Urruty, JB., Strodiot, JJ. & Nguyen, V.H. Generalized Hessian matrix and second-order optimality conditions for problems withC 1,1 data. Appl Math Optim 11, 43–56 (1984). https://doi.org/10.1007/BF01442169
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DOI: https://doi.org/10.1007/BF01442169