Implicit Multifunction Theorems

We prove a general implicit function theorem for multifunctions with a metric estimate on the implicit multifunction and a characterization of its coderivative. Traditional open covering theorems, stability results, and sufficient conditions for a multifunction to be metrically regular or pseudo-Lipschitzian can be deduced from this implicit function theorem. We prove this implicit multifunction theorem by reducing it to an implicit function/solvability theorem for functions. This approach can also be used to prove the Robinson–Ursescu open mapping theorem. As a tool for this alternative proof of the Robinson–Ursescu theorem, we also establish a refined version of the multidirectional mean value inequality which is of independent interest.

[1]  B. N. Pshenichnyi.  Necessary Conditions for an Extremum , 1971 .

[2]  I. Ekeland.  On the variational principle , 1974 .

[3]  C. Ursescu.  Multifunctions with convex closed graph , 1975 .

[4]  F. Clarke,et al.  Topological Geometry: THE INVERSE FUNCTION THEOREM , 1981 .

[5]  S. M. Robinson.  Stability Theory for Systems of Inequalities, Part II: Differentiable Nonlinear Systems , 1976 .

[6]  Stephen M. Robinson,et al.  Regularity and Stability for Convex Multivalued Functions , 1976, Math. Oper. Res..

[7]  Frank H. Clarke,et al.  A New Approach to Lagrange Multipliers , 1976, Math. Oper. Res..

[8]  Steven A. Orszag,et al.  CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS , 1978 .

[9]  A. Ioffe.  Regular points of Lipschitz functions , 1979 .

[10]  J. Aubin.  Contingent Derivatives of Set-Valued Maps and Existence of Solutions to Nonlinear Inclusions and Differential Inclusions. , 1980 .

[11]  J. Warga,et al.  Fat homeomorphisms and unbounded derivate containers , 1981 .

[12]  A. Ioffe.  Nonsmooth analysis: differential calculus of nondifferentiable mappings , 1981 .

[13]  J. Penot.  On regularity conditions in mathematical programming , 1982 .

[14]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[15]  J. Aubin,et al.  On inverse function theorems for set-valued maps , 1984 .

[16]  A. Ioffe.  Calculus of Dini subdifferentials of functions and contingent coderivatives of set-valued maps , 1984 .

[17]  R. Rockafellar,et al.  Lipschitzian properties of multifunctions , 1985 .

[18]  J. Borwein.  Stability and regular points of inequality systems , 1986 .

[19]  Theorems on implicit functions for multivalued mappings , 1986 .

[20]  A. Ioffe.  Global Surjection and Global Inverse Mapping Theorems in Banach Spaces , 1987 .

[21]  A. Ioffe,et al.  On the local surjection property , 1987 .

[22]  H. Frankowska.  An open mapping principle for set-valued maps , 1987 .

[23]  J. Borwein,et al.  A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions , 1987 .

[24]  J. Borwein,et al.  Verifiable necessary and sufficient conditions for openness and regularity of set-valued and single-valued maps , 1988 .

[25]  J. Warga.  Homeomorphisms and local C 1 approximations , 1988 .

[26]  J. Penot.  Metric regularity, openness and Lipschitzian behavior of multifunctions , 1989 .

[27]  A. Ioffe.  Approximate subdifferentials and applications 3: the metric theory , 1989 .

[28]  R. Phelps.  Convex Functions, Monotone Operators and Differentiability , 1989 .

[29]  F. Clarke.  Methods of dynamic and nonsmooth optimization , 1989 .

[30]  H. Frankowska.  Some inverse mapping theorems , 1990 .

[31]  J. Aubin.  Set-valued analysis , 1990 .

[32]  L. Thibault.  On subdifferential of optimal value functions , 1991 .

[33]  Nguyen Dong Yen,et al.  On implicit function theorems for set-valued maps and their application to mathematical programming under inclusion constraints , 1991 .

[34]  Stephen M. Robinson,et al.  An Implicit-Function Theorem for a Class of Nonsmooth Functions , 1991, Math. Oper. Res..

[35]  Implicit Function Theorems and Lagrange Multipliers , 1991 .

[36]  Nguyen Dong Yen,et al.  Correction: On implicit function theorems for set-valued maps and their application to mathematical programming under inclusion constraints , 1992 .

[37]  R. DeVille,et al.  A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions , 1993 .

[38]  B. Mordukhovich.  Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions , 1993 .

[39]  B. Mordukhovich.  Sensitivity analysis for constraint and variational systems by means of set-valued differentiation , 1994 .

[40]  B. Mordukhovich.  Lipschitzian stability of constraint systems and generalized equations , 1994 .

[41]  R. Rockafellar,et al.  Sensitivity analysis of solutions to generalized equations , 1994 .

[42]  B. Mordukhovich.  Generalized Differential Calculus for Nonsmooth and Set-Valued Mappings , 1994 .

[43]  William W. Hager,et al.  An inverse mapping theorem for set-valued maps , 1994 .

[44]  William W. Hager,et al.  Implicit Functions, Lipschitz Maps, and Stability in Optimization , 1994, Math. Oper. Res..

[45]  Frank H. Clarke,et al.  Mean value inequalities , 1994 .

[46]  B. Mordukhovich,et al.  Differential characterizations of covering, metric regularity, and Lispchitzian properties of multifunctions between Banach spaces , 1995 .

[47]  Adam B. Levy,et al.  Implicit multifunction theorems for the sensitivity analysis of variational conditions , 1996, Math. Program..

[48]  Jonathan M. Borwein,et al.  Proximal analysis in smooth spaces , 1996 .

[49]  B. Mordukhovich,et al.  Nonsmooth sequential analysis in Asplund spaces , 1996 .

[50]  Yu. S. Ledyaev,et al.  Optimality conditions for implicit controlled systems , 1996 .

[51]  J. Penot,et al.  SUBDIFFERENTIALS OF PERFORMANCE FUNCTIONS AND CALCULUS OF CODERIVATIVES OF SET-VALUED MAPPINGS , 1996 .

[52]  Jonathan M. Borwein,et al.  Viscosity Solutions and Viscosity Subderivatives in Smooth Banach Spaces with Applications to Metric Regularity , 1996 .

[53]  F. Clarke.  Solving equations by the decrease principle , 1997 .

[54]  B. Mordukhovich,et al.  Stablity of Set-Valued Mappings In Infinite Dimensions: Point Criteria and Applications , 1997 .

[55]  L. Vinet.  Advances in mathematical sciences : CRM's 25 years , 1997 .

[56]  Jean-Paul Penot,et al.  Subtraction Theorems and Approximate Openness for Multifunctions: Topological and Infinitesimal Viewpoints☆ , 1998 .

[57]  Qiji Zhu,et al.  Clarke-Ledyaev mean value inequalities in smooth Banach spaces , 1998 .

[58]  Yu. S. Ledyaev,et al.  Nonsmooth analysis and control theory , 1998 .

[59]  Héctor J. Sussmann,et al.  Multidifferential Calculus: Chain Rule, Open Mapping and Transversal Intersection Theorems , 1998 .

[60]  Qiji J. Zhu,et al.  The Equivalence of Several Basic Theorems for Subdifferentials , 1998 .

[61]  Alexander D. Ioffe,et al.  Fuzzy Principles and Characterization of Trustworthiness , 1998 .