Implicit Functions and Solution Mappings

The purpose of this self-contained work is to provide a reference on the topic and to provide a unified collection of a number of results which are currently scattered throughout the literature. The first chapter of the book treats the classical implicit function theorem in a way that will be useful for students and teachers of undergraduate calculus. The remaining part becomes gradually more advanced, and considers implicit mappings defined by relations other than equations, e.g., variational problems. Applications to numerical analysis and optimization are also provided.

[1]  A General Implicit Function Theorem with an Application to Problems of Relative Minima , 1920 .

[2]  S. Banach.  Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales , 1922 .

[3]  T. H. Hildebrandt,et al.  Implicit functions and their differentials in general analysis , 1927 .

[4]  A. Hoffman.  On approximate solutions of systems of linear inequalities , 1952 .

[5]  E. B. Leach,et al.  A note on inverse function theorems , 1961 .

[6]  R. Tyrrell Rockafellar,et al.  Monotone processes of convex and concave type , 1967 .

[7]  M. J. D. Powell,et al.  Nonlinear Programming—Sequential Unconstrained Minimization Techniques , 1969 .

[8]  S. Nadler.  Multi-valued contraction mappings. , 1969 .

[9]  R. Wets,et al.  A Lipschitzian characterization of convex polyhedra , 1969 .

[10]  Vincent J. Mancuso.  SOME MAPPING THEOREMS , 1971 .

[11]  Albert Nijenhuis.  Strong Derivatives and Inverse Mappings , 1974 .

[12]  Hubert Halkin,et al.  Implicit Functions and Optimization Problems without Continuous Differentiability of the Data , 1974 .

[13]  P. Kenderov,et al.  Semi-continuity of set-valued monotone mappings , 1975 .

[14]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

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

[16]  R. Tyrrell Rockafellar,et al.  Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming , 1976, Math. Oper. Res..

[17]  C. Berge.  Topological Spaces: including a treatment of multi-valued functions , 2010 .

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

[19]  P. Huard,et al.  Point-to-set maps and mathematical programming , 1979 .

[20]  Stephen M. Robinson,et al.  Strongly Regular Generalized Equations , 1980, Math. Oper. Res..

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

[22]  M. Kojima.  Strongly Stable Stationary Solutions in Nonlinear Programs. , 1980 .

[23]  S. M. Robinson.  Some continuity properties of polyhedral multifunctions , 1981 .

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

[25]  Richard S. Hamilton,et al.  The inverse function theorem of Nash and Moser , 1982 .

[26]  Jonathan M. Borwein,et al.  Adjoint Process Duality , 1983, Math. Oper. Res..

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

[28]  S. M. Robinson.  Local Structure of Feasible Sets in Nonlinear Programming - Part II. Nondegeneracy , 1984 .

[29]  Jean-Pierre Aubin,et al.  Lipschitz Behavior of Solutions to Convex Minimization Problems , 1984, Math. Oper. Res..

[30]  J. Aubin,et al.  Applied Nonlinear Analysis , 1984 .

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

[32]  Jan-J. Rückmann,et al.  On inertia and schur complement in optimization , 1987 .

[33]  R. Rockafellar.  Proto-Differentiability of Set-Valued Mappings and its Applications in Optimization☆ , 1989 .

[34]  W. Alt.  The lagrange-newton method for infinite-dimensional optimization problems , 1990 .

[35]  V. N. Bogaevski,et al.  Matrix Perturbation Theory , 1991 .

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

[37]  R. Tyrrell Rockafellar,et al.  Sensitivity analysis for nonsmooth generalized equations , 1992, Math. Program..

[38]  Stephen M. Robinson,et al.  Normal Maps Induced by Linear Transformations , 1992, Math. Oper. Res..

[39]  W. Hager,et al.  Lipschitzian stability in nonlinear control and optimization , 1993 .

[40]  R. DeVille,et al.  Smoothness and renormings in Banach spaces , 1993 .

[41]  D. Ralph.  A new proof of Robinson's homeomorphism theorem for pl-normal maps , 1993 .

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

[43]  Robert Gardner,et al.  Introduction To Real Analysis , 1994 .

[44]  Wu Li.  Sharp Lipschitz Constants for Basic Optimal Solutions and Basic Feasible Solutions of Linear Programs , 1994 .

[45]  E. Levitin.  Perturbation Theory in Mathematical Programming and Its Applications , 1994 .

[46]  S. M. Robinson.  Newton's method for a class of nonsmooth functions , 1994 .

[47]  E. Zeidler.  The Implicit Function Theorem , 1995 .

[48]  Asen L. Dontchev,et al.  Characterizations of Lipschitz Stability in Optimization , 1995 .

[49]  Asen L. Dontchev,et al.  Implicit function theorems for generalized equations , 1995, Math. Program..

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

[51]  Heinz H. Bauschke,et al.  On Projection Algorithms for Solving Convex Feasibility Problems , 1996, SIAM Rev..

[52]  R. Tyrrell Rockafellar,et al.  Characterizations of Strong Regularity for Variational Inequalities over Polyhedral Convex Sets , 1996, SIAM J. Optim..

[53]  Asen L. Dontchev,et al.  An a priori Estimate for Discrete Approximations in Nonlinear Optimal Control , 1996 .

[54]  Zsolt Páles,et al.  Inverse and Implicit Function Theorems for Nonsmooth Maps in Banach Spaces , 1997 .

[55]  R. Poliquin,et al.  Characterizing the Single-Valuedness of Multifunctions , 1997 .

[56]  Jong-Shi Pang,et al.  Error bounds in mathematical programming , 1997, Math. Program..

[57]  R. Rockafellar,et al.  Characterizations of Lipschitzian Stability in Nonlinear Programming , 2020 .

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

[59]  Yu. S. Ledyaev,et al.  Implicit Multifunction Theorems , 1998 .

[60]  Asen L. Dontchev.  Lipschitzian Stability of Newton's Method for Variational Inclusions , 1999, System Modelling and Optimization.

[61]  A. S. Lewis,et al.  Ill-Conditioned Convex Processes and Conic Linear Systems , 1999, Math. Oper. Res..

[62]  A. Ioffe.  Metric regularity and subdifferential calculus , 2000 .

[63]  J. Borwein,et al.  Convex Analysis And Nonlinear Optimization , 2000 .

[64]  Yu. S. Ledyaev,et al.  Dualization of the Euler and Hamiltonian inclusions , 2001 .

[65]  Kazimierz Malanowski,et al.  Stability and sensitivity analysis for optimal control problems with control-state constraints , 2001 .

[66]  Allan P. Donsig,et al.  Real Analysis with Real Applications , 2001 .

[67]  R. Tyrrell Rockafellar,et al.  Ample Parameterization of Variational Inclusions , 2001, SIAM J. Optim..

[68]  R. Rockafellar,et al.  The radius of metric regularity , 2002 .

[69]  C. Zălinescu.  Convex analysis in general vector spaces , 2002 .

[70]  Jonathan M. Borwein,et al.  On the Bartle-Graves theorem , 2003 .

[71]  Alexander D. Ioffe.  On robustness of the regularity property of maps , 2003 .

[72]  A. D. Ioffe.  On stability estimates for the regularity property of maps , 2003 .

[73]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[74]  Asen L. Dontchev.  A Local Selection Theorem for Metrically Regular Mappings , 2004 .

[75]  Asen L. Dontchev,et al.  Regularity and Conditioning of Solution Mappings in Variational Analysis , 2004 .

[76]  Vladimir M. Veliov.  Approximations with Error Estimates for Optimal Control Problems for Linear Systems , 2005, LSSC.

[77]  Giuseppe Buttazzo,et al.  Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization (Mps-Siam Series on Optimization 6) , 2005 .

[78]  J. Borwein,et al.  Techniques of variational analysis , 2005 .

[79]  Diethard Klatte,et al.  Nonsmooth Equations in Optimization: "Regularity, Calculus, Methods And Applications" , 2006 .

[80]  F. Giannessi.  Variational Analysis and Generalized Differentiation , 2006 .

[81]  A. Dontchev,et al.  Convergence of the proximal point method for metrically regular mappings , 2007 .

[82]  Aubin Criterion for Metric Regularity∗ , 2006 .

[83]  S. M. Robinson,et al.  Solution Continuity in Monotone Affine Variational Inequalities , 2007, SIAM J. Optim..

[84]  Aram V. Arutyunov,et al.  Covering mappings in metric spaces and fixed points , 2007 .

[85]  Hisayoshi Kunimune,et al.  Open Mapping Theorem , 2008, Formaliz. Math..

[86]  R. Rockafellar,et al.  Implicit Functions and Solution Mappings: A View from Variational Analysis , 2009 .

[87]  Richard W. Cottle,et al.  Linear Complementarity Problem , 2009, Encyclopedia of Optimization.

[88]  rha 4. Mdha.  Lipschitzian Properties of Multifunctions , 2010 .

[89]  J. Borwein,et al.  Convex Functions: Constructions, Characterizations and Counterexamples , 2010 .

[90]  Philipp Birken,et al.  Numerical Linear Algebra , 2011, Encyclopedia of Parallel Computing.