Abstract
Hiriart-Urruty and the author recently introduced the notions of Dupin indicatrices for nonsmooth convex surfaces and studied them in connection with their concept of a second-order subdifferential for convex functions. They noticed that second-order subdifferentials can be viewed as limit sets of difference quotients involving approximate subdifferentials. In this paper, we elaborate this point in a more detailed way and discuss some related questions.
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Hiriart-Urruty, J. B., Strodiot, J. J., andNguyen, V. H.,Generalized Hessian Matrix and Second-Order Conditions for Problems with C 1,1 Data, Applied Mathematics and Optimization, Vol. 11, pp. 43–56, 1984.
Cominetti, R., andCorrea, R.,Sur une Dérivée du Second Ordre en Analyse Non-Différentiable, Comptes Rendus de l'Académie des Sciences de Paris, Vol. 303, Série I, pp. 861–864, 1986.
Cominetti, R., andCorrea, R.,A Generalized Second-Order Derivative in Nonsmooth Optimization, SIAM Journal on Control and Optimization, Vol. 28, pp. 789–809, 1990.
Haussmann, U. G.,A Probabilistic Approach to the Generalized Hessian, Preprint, Department of Mathematics, University of British Columbia, 1989.
Ponstein, J.,On Generalized Higher-Order Derivatives, Preprint, Institute of Economics, University of Gröningen, 1985.
Bedelbaev, A. A.,Necessary and Sufficient Second-Order Optimality Conditions in the Discrete Bolza Problem of Optimal Control, Izvestiia Akademiia Nauk Kazakhskoi SSR, Seriia Fiziko-Matematicheskaia, No. 1, pp. 5–11, 1980 (in Russian).
Hiriart-Urruty, J. B.,A New Set-Valued Second-Order Derivative for Convex Functions, Mathematics for Optimization, Edited by J. B. Hiriart-Urruty, North-Holland, Amsterdam, Holland, pp. 157–182, 1986.
Seeger, A.,Analyse du Second Ordre de Problèmes Non-Différentiables, PhD Thesis, Laboratoire d'Analyse Numérique, Université Paul Sabatier, Toulouse, 1986.
Aubin, J. P.,Comportement Lipschitzien des Solutions de Problèmes de Minimisation Convexes, Comptes Rendus de l'Académie des Sciences de Paris, Vol. 295, Série I, pp. 235–238, 1982.
Aubin, J. P.,Lipschitz Behavior of Solutions to Convex Minimization Problems, Mathematics of Operations Research, Vol. 9, pp. 87–111, 1984.
Aubin, J. P., andEkeland, I.,Applied Nonlinear Analysis, J. Wiley and Sons, New York, New York, 1984.
Rockafellar, R. T.,Generalized Second Derivatives of Convex Functions and Saddle Functions, Transactions of the American Mathematical Society, Vol. 322, pp. 51–77, 1991.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
Lemaréchal, C.,Constructing Bundle Methods for Convex Optimization, Mathematics for Optimization, Edited by J. B. Hiriart-Urruty, North-Holland, Amsterdam, Holland, pp. 201–240, 1986.
Zowe, J.,Nodifferentiable Optimization—A Motivation and a Short Introduction into the Subgradient and the Bundle Concept, Computational Mathematical Programming, Edited by K. Schittkowski, Springer-Verlag, Berlin, Germany, 1985.
Hiriart-Urruty, J. B.,Limiting Behavior of the Approximate First-Order and Second-Order Directional Derivatives for a Convex Function, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 6, pp. 1309–1326, 1982.
Hiriart-Urruty, J. B.,Approximating a Second-Order Directional Derivative for Nonsmooth Convex Functions, SIAM Journal on Control and Optimization, Vol. 20, pp. 783–807, 1982.
Hiriart-Urruty, J. B.,The Approximate First-Order and Second-Order Directional Derivatives for a Convex Function, Proceedings of the Conference on Mathematical Theories of Optimization, S. Margherita Ligure, Italy, 1981; Lecture Notes in Mathematics, Springer-Verlag, Berlin, Germany, Vol. 979, pp. 144–177, 1983.
Hiriart-Urruty, J. B., andSeeger, A.,The Second-Order Subdifferential and the Dupin Indicatrices of a Non-Differentiable Convex Function, Proceedings of the London Mathematical Society, Vol. 58, pp. 351–365, 1989.
Lemaréchal, C., andZowe, J.,Some Remarks on the Construction of Higher-Order Algorithms in Convex Optimization, Applied Mathematics and Optimization, Vol. 10, pp. 51–68, 1983.
Attouch, H.,Variational Convergence for Functions and Operators, Pitman, London, England, 1984.
Rockafellar, R. T.,Maximal Monotone Relations and the Second-Derivatives of Nonsmooth Functions, Annales de l'Institut Henri Poincaré, Vol. 2, pp. 167–184, 1985.
Ndoutoume, J. L.,Calcul Différentiel du Second Ordre, Publications AVAMAC, University of Perpignan, Perpignan, France, 1987.
Do, C. H.,Second-Order Nonsmooth Analysis and Sensitivity in Optimization Problems Involving Convex Integral Functionals, PhD Thesis, Department of Mathematics, University of Washington, Seattle, 1989.
Mosco, U.,On the Continuity of the Young-Fenchel Transform, Journal of Mathematical Analysis and Applications, Vol. 35, pp. 518–535, 1971.
Busemann, H.,Convex Surfaces, Interscience (John Wiley), New York, New York, 1958.
Schneider, R.,Boundary Structures and Curvatures of Convex Bodies, Proceedings of the Geometry Symposium, Siegen, Germany, 1978; Birkhäuser, Basel, Switzerland, 1979.
Hiriart-Urruty, J. B., andSeeger, A.,Règles de Calcul sur le Sous-Différentiel Second d'une Fonction Convexe, Comptes Rendus de l'Académie des Sciences de Paris, Vol. 304, Série I, pp. 259–262, 1987.
Hiriart-Urruty, J. B., andSeeger, A.,Calculus Rules on a New Set-Valued Second-Order Derivative for Convex Functions, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 13, pp. 721–738, 1989.
Rockafellar, R. T.,First- and Second-Order Epi-Differentiability in Nonlinear Programming, Transactions of the American Mathematical Society, Vol. 307, pp. 75–108, 1988.
Rockafellar, R. T.,Second-Order Optimality Conditions in Nonlinear Programming Obtained by Way of Epi-Derivatives, Mathematics of Operations Research, Vol. 14, pp. 462–484, 1989.
Poliquin, R.,Proto-Differentiation and Integration of Proximal Subgradients, PhD Thesis, Department of Mathematics, University of Washington, Seattle, 1988.
Poliquin, R.,Proto-Differentiation of Subgradient Set-Valued Mappings, Canadian Journal of Mathematics, Vol. 42, pp. 520–532, 1990.
Seeger, A.,Second Derivatives of a Convex Function and of Its Legendre-Fenchel Transformate, Preprint, Department of Mathematics, University of Washington, Seattle, 1989.
Seeger, A.,Complément de Schur et Sous-Différentiel du Second Ordre d'une Fonction Convexe, Aequationes Mathematicae, Vol. 42, pp. 47–71, 1991.
Hiriart-Urruty, J. B.,ɛ-Subdifferential Calculus, Research Notes in Mathematics, Pitman, London, England, Vol. 57, pp. 43–92, 1982.
Kutateladze, S. S.,Convex ɛ-Programming, Soviet Mathematics Doklady, Vol. 20, pp. 391–393, 1979.
Hiriart-Urruty, J. B.,From Convex Optimization to Nonconvex Optimization, Nonsmooth Optimization and Related Topics, Edited by F. H. Clarke, V. H. Demyanov, and F. Giannessi, Plenum Press, New York, New York, pp. 219–239, 1989.
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Communicated by A. V. Fiacco
The author is grateful to the referees for their helpful comments.
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Seeger, A. Limiting behavior of the approximate second-order subdifferential of a convex function. J Optim Theory Appl 74, 527–544 (1992). https://doi.org/10.1007/BF00940325
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DOI: https://doi.org/10.1007/BF00940325