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Quantitative stability of variational systems. I. The epigraphical distance

Authors: Hédy Attouch and Roger J.-B. Wets
Journal: Trans. Amer. Math. Soc. 328 (1991), 695-729
MSC: Primary 90C31; Secondary 49J52, 54C35
MathSciNet review: 1018570
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Abstract: This paper proposes a global measure for the distance between the elements of a variational system (parametrized families of optimization problems).

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  • [1] H. Attouch, Variational convergence for functions and operators, Appl. Math. Ser., Pitman, London, 1984. MR 773850 (86f:49002)
  • [2] H. Attouch, D. Azé, and J.-C. Peralba, Comparaison de différentes métriques liées à l'épiconvergence (in preparation).
  • [3] H. Attouch, D. Azé and R. J.-B. Wets, On the continuity properties of the partial Legendre-Fenchel transform: convergence of sequences of augmented lagrangian functions, Moreau-Yosida approximates and subdifferential operators, Optimization Days 1985, Proc. Journées Fer=(J.-B. Hiriart-Urruty, ed.), North-Holland, Amsterdam, 1986. MR 874359 (88d:90086)
  • [4] H. Attouch and A. Damlamian, Problèmes d'évolution dans les ilubert et applications, J. Math. Pures Appl. 54 (1975), 53-74. MR 0454756 (56:13004)
  • [5] H. Attouch, R. Lucchetti, and R. J.-B. Wets, The topology of the $ \rho $-Hausdorff distance, Ann. Mat. Pura Appl. Ann. Mat. Pura Appl. (to appear). MR 1163212 (93d:54022)
  • [6] H. Attouch and R. J.-B. Wets. Isometries for the Legendre-Fenchel transform, Trans. Amer. Math. Soc. 296 (1986), 33-60. MR 837797 (87k:49023)
  • [7] -, Another isometr ffor the Legendre-Fenchel transform, J. Math. Anal. Appl. 131 (1988), 404-411. MR 935277 (89e:44002)
  • [8] -, Epigraphical analysis, Analyse Non Linéaire (H. Attouch, J.-P. Aubin, F. Clarke, and I. Ekeland, eds.), Gauthier-Villars, Paris, 1989, pp. 73-100. MR 1204007 (93h:49003)
  • [9] -, Quantitative stability of variational systems: II. A framework for nonlinear conditioning, IIASA Working Paper 88-9, Laxemburg, Austria, February 1988.
  • [10] -, Lipschitzian stability of the $ \varepsilon $-approximate solutions in convex optimization, IIASA Working Paper WP-87-25, Laxemburg, Austria, March 1987.
  • [11] J.-P. Aubin, Comportement lipschitzien des solutions de problèmes de minimisation convexes, C. R. Acad. Sci. Paris 295 (1982), 235-238. MR 681586 (84b:90076)
  • [12] D. Azé, Convergence variationlile et dualité. Applications en calcul des variations et en programmation mathématique, Thèse de Doctorat, AVAMAC-Perpignan, 1985.
  • [13] D. Azé and J.-P. Penot, Operations on convergent families of sets and functions, Technical Report AVAMAC, Perpignan, 1987.
  • [14] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de ilubert, North-Holland, Amsterdam, 1973.
  • [15] V. Chiado Piat, G. Dal Maso, and A. Defranceschi, $ G$-convergence of monotone operators, SISSA Ref. 84M, Trieste, July 1988.
  • [16] N. Kenmochi, The semi-discretization method and time dependent parabolic variational inequalities, Proc. Japan Acad. Ser. A Math. Sci. 50 (1974), 714-717. MR 0375020 (51:11216)
  • [17] J.-J. Moreau, Intersection of moving convex sets in a normed space, Math. Scand. 36 (1975), 159-173. MR 0442644 (56:1025)
  • [18] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. in Math. 3 (1969). MR 0298508 (45:7560)
  • [19] R. T. Rockafellar, Maximal monotone relations and the second derivative of nonsmooth functions, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), 510-585. MR 797269 (87c:49021)
  • [20] R. T. Rockafellar and R. J.-B. Wets, Variational systems, an introduction, Multifunctions and Integrands: Stochastic Analysis, Approximation and Optimization (G. Salinetti, ed.) Lecture Notes in Math., vol. 1091, Springer-Verlag, Berlin, 1984, pp. 1-54. MR 785574 (86h:90116)
  • [21] G.. Salinetti and R. J.-B. Wets, On the convergence of sequence of convex sets i nfinite dimensions, SIAM Rev. 21 (1979), 16-33. MR 516381 (80h:52007)
  • [22] -, On the convergence of closed-valued measurable multifunctions, Trans. Amer. Math. Soc. 266 (1981), 275-289. MR 613796 (82k:28007)
  • [23] R. Schultz, Estimates for Kuhn-Tucker points of perturbed convex programs, Technical Report, Humboldt Univ., Berlin, 1986.
  • [24] D. Walkup and R. J.-B. Wets, Continuity of some convex cone-valued mappings, Proc. Amer. Math. Soc. 18 (1967), 229-235. MR 0209806 (35:702)
  • [25] R. J.-B. Wets, Convergence of convex functions, variational inequalities and convex optimization problems, Variational Inequalities and Complementarity Problems (R. Cottle, F. Giannessi, and J. L. Lions, eds.), Wiley, Chichester, 1980, pp. 405-419. MR 578760 (83a:90140)

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Article copyright: © Copyright 1991 American Mathematical Society

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