Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis
HTML articles powered by AMS MathViewer
- by Boris Mordukhovich PDF
- Trans. Amer. Math. Soc. 343 (1994), 609-657 Request permission
Abstract:
In this paper we develop a stability theory for broad classes of parametric generalized equations and variational inequalities in finite dimensions. These objects have a wide range of applications in optimization, nonlinear analysis, mathematical economics, etc. Our main concern is Lipschitzian stability of multivalued solution maps depending on parameters. We employ a new approach of nonsmooth analysis based on the generalized differentiation of multivalued and nonsmooth operators. This approach allows us to obtain effective sufficient conditions as well as necessary and sufficient conditions for a natural Lipschitzian behavior of solution maps. In particular, we prove new criteria for the existence of Lipschitzian multivalued and single-valued implicit functions.References
- Jean-Pierre Aubin, Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9 (1984), no. 1, 87–111. MR 736641, DOI 10.1287/moor.9.1.87
- Jean-Pierre Aubin and Hélène Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications, vol. 2, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1048347
- J. M. Borwein and D. M. Zhuang, Verifiable necessary and sufficient conditions for openness and regularity of set-valued and single-valued maps, J. Math. Anal. Appl. 134 (1988), no. 2, 441–459. MR 961349, DOI 10.1016/0022-247X(88)90034-0
- Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 709590
- Michael G. Crandall and Pierre-Louis Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), no. 1, 1–42. MR 690039, DOI 10.1090/S0002-9947-1983-0690039-8
- Asen L. Dontchev and William W. Hager, Implicit functions, Lipschitz maps, and stability in optimization, Math. Oper. Res. 19 (1994), no. 3, 753–768. MR 1288898, DOI 10.1287/moor.19.3.753 M. S. Gowda and J.-S. Pang, Stability analysis of variational inequality and nonlinear complementarity problems, via the mixed linear complementarity problem and degree theory, Math. Oper. Res. (to appear).
- Patrick T. Harker and Jong-Shi Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Math. Programming 48 (1990), no. 2, (Ser. B), 161–220. MR 1073707, DOI 10.1007/BF01582255
- A. D. Ioffe, Approximate subdifferentials and applications. I. The finite-dimensional theory, Trans. Amer. Math. Soc. 281 (1984), no. 1, 389–416. MR 719677, DOI 10.1090/S0002-9947-1984-0719677-1
- Alan J. King and R. Tyrrell Rockafellar, Sensitivity analysis for nonsmooth generalized equations, Math. Programming 55 (1992), no. 2, Ser. A, 193–212. MR 1167597, DOI 10.1007/BF01581199 A. Kruger and B. Mordukhovich, Generalized normals and derivatives, and necessary conditions for extrema in nondifferentiable programming. I, Depon. VINITI No. 408-80 (1980, Russian).
- A. Ja. Kruger and B. Š. Morduhovič, Extremal points and the Euler equation in nonsmooth optimization problems, Dokl. Akad. Nauk BSSR 24 (1980), no. 8, 684–687, 763 (Russian, with English summary). MR 587714
- Bernd Kummer, An implicit-function theorem for $C^{0,1}$-equations and parametric $C^{1,1}$-optimization, J. Math. Anal. Appl. 158 (1991), no. 1, 35–46. MR 1113397, DOI 10.1016/0022-247X(91)90264-Z
- B. Kummer, Lipschitzian inverse functions, directional derivatives, and applications in $C^{1,1}$ optimization, J. Optim. Theory Appl. 70 (1991), no. 3, 561–581. MR 1124778, DOI 10.1007/BF00941302
- Jerzy Kyparisis, Parametric variational inequalities with multivalued solution sets, Math. Oper. Res. 17 (1992), no. 2, 341–364. MR 1161159, DOI 10.1287/moor.17.2.341 J. Liu, Nonsingular solutions of finite-dimensional variational inequalities: theory and methods, preprint, 1992.
- B. Sh. Mordukhovich, Maximum principle in the problem of time optimal response with nonsmooth constraints, Prikl. Mat. Meh. 40 (1976), no. 6, 1014–1023 (Russian); English transl., J. Appl. Math. Mech. 40 (1976), no. 6, 960–969 (1977). MR 0487669, DOI 10.1016/0021-8928(76)90136-2 —, Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems, Soviet Math. Dokl. 22 (1980), 526-530.
- B. Sh. Mordukhovich, Metody approksimatsiĭ v zadachakh optimizatsii i upravleniya, “Nauka”, Moscow, 1988 (Russian). MR 945143
- Boris S. Mordukhovich, Sensitivity analysis in nonsmooth optimization, Theoretical aspects of industrial design (Wright-Patterson Air Force Base, OH, 1990) SIAM, Philadelphia, PA, 1992, pp. 32–46. MR 1157413
- B. S. Mordukhovich, On variational analysis of differential inclusions, Optimization and nonlinear analysis (Haifa, 1990) Pitman Res. Notes Math. Ser., vol. 244, Longman Sci. Tech., Harlow, 1992, pp. 199–213. MR 1184644
- Boris Mordukhovich, Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions, Trans. Amer. Math. Soc. 340 (1993), no. 1, 1–35. MR 1156300, DOI 10.1090/S0002-9947-1993-1156300-4
- Boris Mordukhovich, Lipschitzian stability of constraint systems and generalized equations, Nonlinear Anal. 22 (1994), no. 2, 173–206. MR 1258955, DOI 10.1016/0362-546X(94)90033-7
- Jong-Shi Pang, A degree-theoretic approach to parametric nonsmooth equations with multivalued perturbed solution sets, Math. Programming 62 (1993), no. 2, Ser. B, 359–383. MR 1247621, DOI 10.1007/BF01585174
- Jean-Paul Penot, Metric regularity, openness and Lipschitzian behavior of multifunctions, Nonlinear Anal. 13 (1989), no. 6, 629–643. MR 998509, DOI 10.1016/0362-546X(89)90083-7
- Yuping Qiu and Thomas L. Magnanti, Sensitivity analysis for variational inequalities, Math. Oper. Res. 17 (1992), no. 1, 61–76. MR 1148778, DOI 10.1287/moor.17.1.61
- Stephen M. Robinson, Stability theory for systems of inequalities. II. Differentiable nonlinear systems, SIAM J. Numer. Anal. 13 (1976), no. 4, 497–513. MR 410522, DOI 10.1137/0713043 —, Generalized equations and their solutions, part I: basic theory, Math. Programming Stud. 10 (1979), 128-141.
- Stephen M. Robinson, Strongly regular generalized equations, Math. Oper. Res. 5 (1980), no. 1, 43–62. MR 561153, DOI 10.1287/moor.5.1.43 —, Generalized equations, Math, programming: The State of Art (A. Bachem, M. Grötschel, and B. Korte, eds.), Springer-Verlag, Berlin and New York, 1982, pp. 346-367.
- Stephen M. Robinson, An implicit-function theorem for a class of nonsmooth functions, Math. Oper. Res. 16 (1991), no. 2, 292–309. MR 1106803, DOI 10.1287/moor.16.2.292
- R. T. Rockafellar, Conjugate convex functions in optimal control and the calculus of variations, J. Math. Anal. Appl. 32 (1970), 174–222. MR 266020, DOI 10.1016/0022-247X(70)90324-0
- Ralph T. Rockafellar, The theory of subgradients and its applications to problems of optimization, R & E, vol. 1, Heldermann Verlag, Berlin, 1981. Convex and nonconvex functions. MR 623763
- R. Tyrrell Rockafellar, Lipschitzian properties of multifunctions, Nonlinear Anal. 9 (1985), no. 8, 867–885. MR 799890, DOI 10.1016/0362-546X(85)90024-0
- R. T. Rockafellar, Maximal monotone relations and the second derivatives of nonsmooth functions, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 3, 167–184 (English, with French summary). MR 797269
- R. Tyrrell Rockafellar and Roger J.-B. Wets, Variational analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317, Springer-Verlag, Berlin, 1998. MR 1491362, DOI 10.1007/978-3-642-02431-3 A. Shapiro, Sensitivity analysis of parametric programs via generalized equations, preprint, 1992.
- Lionel Thibault, Subdifferentials of compactly Lipschitzian vector-valued functions, Ann. Mat. Pura Appl. (4) 125 (1980), 157–192. MR 605208, DOI 10.1007/BF01789411
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 343 (1994), 609-657
- MSC: Primary 49J52; Secondary 49K40, 90C31
- DOI: https://doi.org/10.1090/S0002-9947-1994-1242786-4
- MathSciNet review: 1242786