The extremal principle and its applications to optimization and economics

This paper is devoted to the extremal principle in variational analysis that can be viewed as a variational analog of the classical convex separation principle in nonconvex settings. We consider two basic versions of the extremal principle in Banach spaces formulated in terms of Frechet-like normals and their sequential limits. Then we discuss various applications of these results to the generalized differential calculus for nonsmooth and set-valued mappings, to necessary optimality and suboptimality conditions in problems of constrained optimization, and to the study of Pareto optimality in nonconvex models of welfare economics.

[1]  A. Kruger.  Properties of generalized differentials , 1985 .

[2]  A. Ioffe,et al.  ON SUBDIFFERENTIABILITY SPACES , 1983 .

[3]  A. Kruger,et al.  Generalized differentials of nonsmooth functions, and necessary conditions for an extremum , 1985 .

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

[5]  G. Debreu,et al.  Theory of Value , 1959 .

[6]  Boris S. Mordukhovich,et al.  Coderivatives of set-valued mappings: Calculus and applications , 1997 .

[7]  Marc Lassonde,et al.  First-order rules for nonsmooth constrained optimization , 2001 .

[8]  Rajiv Vohra,et al.  Pareto optimal allocations of nonconvex economies in locally convex spaces , 1988 .

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

[10]  Roger Guesnerie,et al.  Pareto Optimality in Non-Convex Economies , 1975 .

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

[12]  Jonathan M. Borwein,et al.  Necessary conditions for constrained optimization problems with semicontinuous and continuous data , 1998 .

[13]  R. Rockafellar.  Directionally Lipschitzian Functions and Subdifferential Calculus , 1979 .

[14]  J. Borwein,et al.  On the Equivalence of Some Basic Principles in Variational Analysis , 1999 .

[15]  Boris S. Mordukhovich,et al.  Nonconvex differential calculus for infinite-dimensional multifunctions , 1996 .

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

[17]  Boris S. Mordukhovich,et al.  Separable Reduction and Supporting Properties of Fréchet-Like Normals in Banach Spaces , 1999 .

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

[19]  BORIS S. MORDUKHOVICH,et al.  Necessary Suboptimality and Optimality Conditions via Variational Principles , 2002, SIAM J. Control. Optim..

[20]  Boris S. Mordukhovich,et al.  Fuzzy calculus for coderivatives of multifunctions , 1997 .

[21]  Bernard Cornet,et al.  Valuation equilibrium and Pareto optimum in non-convex economies , 1988 .

[22]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[23]  K. Arrow,et al.  EXISTENCE OF AN EQUILIBRIUM FOR A COMPETITIVE ECONOMY , 1954 .

[24]  Boris S. Mordukhovich,et al.  Separable reduction and extremal principles in variational analysis , 2002 .

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

[26]  Makhon ṭekhnologi le-Yiśraʾel Ṭekhniyon,et al.  Optimization and Nonlinear Analysis , 1992 .

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

[28]  Peter Nash.  Systems modelling and optimization , 1981 .

[29]  B. Mordukhovich,et al.  Extremal characterizations of asplund spaces , 1996 .

[30]  Lionel Thibault,et al.  Extensions Of Subdifferential Calculus Rules in Banach Spaces , 1996, Canadian Journal of Mathematics.

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

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

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

[34]  D. Varberg.  Convex Functions , 1973 .

[35]  E. Asplund.  Fréchet differentiability of convex functions , 1968 .

[36]  A.Ya. Dubovitskii,et al.  Extremum problems in the presence of restrictions , 1965 .

[37]  Marián Fabian.  Subdifferentials, Local ε‐Supports and Asplund Spaces , 1986 .

[38]  L. Thibault,et al.  Qualification Conditions for Calculus Rules of Coderivatives of Multivalued Mappings , 1998 .

[39]  Jonathan M. Borwein.  Epi-lipschitz-like sets in banach space: theorems and examples , 1987 .

[40]  B. Mordukhovich.  Maximum principle in the problem of time optimal response with nonsmooth constraints PMM vol. 40, n≗ 6, 1976, pp. 1014-1023 , 1976 .

[41]  Boris S. Mordukhovich,et al.  Pareto Optimality in Nonconvex Economies with Infinite-dimensional Commodity Spaces , 2001, J. Glob. Optim..

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

[43]  J. Borwein,et al.  Survey of subdifferential calculus with applications , 1999 .

[44]  M. Ali Khan.  The Mordukhovich Normal Cone and the Foundations of Welfare Economics , 1999 .

[45]  Marián Fabian,et al.  Sub differentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss , 1989 .

[46]  Jean-Paul Penot,et al.  Compactness Properties, Openness Criteria and Coderivatives , 1998 .

[47]  Boris S. Mordukhovich,et al.  An abstract extremal principle with applications to welfare economics , 2000 .

[48]  B. S. Mordukhovich,et al.  Mixed Coderivatives of Set–Valued Mappings in Variational Analysis , 1998 .

[49]  Bernard Cornet.  Marginal cost pricing and Pareto optimality , 1988 .

[50]  Rajiv Vohra,et al.  An approximate decentralization of Pareto optimal allocations in locally convex spaces , 1988 .

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

[52]  R. Rockafellar,et al.  Clarke's tangent cones and the boundaries of closed sets in Rn , 1979 .

[53]  Boris S. Mordukhovich,et al.  Compactly epi-Lipschitzian convex sets and functions in normed spaces , 2000 .

[54]  Boris S. Mordukhovich,et al.  Nonsmooth Characterizations of Asplund Spaces and Smooth Variational Principles , 1998 .

[55]  Bernard Cornet,et al.  The Second Welfare Theorem in Nonconvex Economies , 1986 .

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

[57]  L. Thibault,et al.  Coderivatives of multivalued mappings, locally compact cones and metric regularity , 1999 .

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

[59]  L. Thibault,et al.  Metric regularity for strongly compactly Lipschitzian mappings , 1995 .

[60]  Jonathan M. Borwein,et al.  A survey of subdifferential calculus with applications , 2002 .

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

[62]  Jonathan M. Borwein,et al.  A nonconvex separation property in Banach spaces , 1998, Math. Methods Oper. Res..