Generalized Convex Duality and its Economic Applicatons

This article presents an approach to generalized convex duality theory based on Fenchel-Moreau conjugations; in particular, it discusses quasiconvex conjugation and duality in detail. It also describes the related topic of microeconomics duality and analyzes the monotonicity of demand functions.

[1]  Richard R Egudo,et al.  Efficiency and generalized convex duality for multiobjective programs , 1989 .

[2]  Juan Enrique Martínez-Legaz,et al.  Duality For D.C. Optimization Over Compact Sets , 2001 .

[3]  J. Martínez-Legaz Generalized Conjugation and Related Topics , 1990 .

[4]  E. N. Barron,et al.  Hopf–Lax-Type Formula forut+H(u, Du)=0 , 1996 .

[5]  œ-conjugation and nonconvex optimization. a survey (part i) , 1983 .

[6]  Do Sang Kim,et al.  Duality For Multiobjective Fractional Variational Problems With Generalized Invexity , 1998 .

[7]  Juan Enrique Martínez-Legaz,et al.  On φ-convexity of convex functions , 1998 .

[8]  Karl-Heinz Elster,et al.  Zur theorie der polarfunktionale , 1974 .

[9]  M. Volle,et al.  Duality in DC Programming , 1998 .

[10]  J.-P. Crouzeix Duality between direct and indirect utility functions: Differentiability properties , 1983 .

[11]  Do Sang Kim,et al.  Optimality and Duality for Invex Nonsmooth Multiobjective programming problems , 2004 .

[12]  Santanu K. Mishra,et al.  Second order symmetric duality in mathematical programming with F-convexity , 2000, Eur. J. Oper. Res..

[13]  Ivan. Singer Some Relations Between Linear Mappings and Conjugations in Idempotent Analysis , 2001 .

[14]  J. Frenk,et al.  Duality theory for convex/quasiconvex functions and its application to optimization , 1994 .

[15]  A. Seeger Recent advances in optimization , 2006 .

[16]  Conditions initiales quasiconvexes dans les équations de Hamilton-Jacobi , 1997 .

[17]  Juan Enrique Martínez-Legaz,et al.  Simplified Global Optimality Conditions in Generalized Conjugation Theory , 1998 .

[18]  Werner Hildenbrand,et al.  Market Demand: Theory and Empirical Evidence , 1994 .

[19]  A. Mas-Colell The Theory Of General Economic Equilibrium , 1985 .

[20]  Nicolas Hadjisavvas,et al.  On generalized cyclically monotone operators and proper quasimonotonictty , 2000 .

[21]  A. Cambini,et al.  Generalized convexity and fractional programming with economic applications : proceedings of the International Workshop on "Generalized Concavity, Fractional Programming, and Economic Applications" held at the University of Pisa, Italy, May 30-June 1, 1988 , 1990 .

[22]  Phan Thien Thach Global optimality criterion and a duality with a zero gap in nonconvex optimization , 1993 .

[23]  Manuel S. Santos,et al.  Duality between direct and indirect preferences , 1993 .

[24]  I. Singer Abstract Convex Analysis , 1997 .

[25]  Ivan Singer Surrogate dual problems and surrogate lagrangians , 1984 .

[26]  D. Primont,et al.  Multi-Output Production and Duality: Theory and Applications , 1994 .

[27]  U. PASSY,et al.  Conjugacy in quasi-convex programming , 1984, Math. Program..

[28]  Ury Passy,et al.  A convex-like duality scheme for quasi-convex programs , 1985, Math. Program..

[29]  B. Brunt The calculus of variations , 2003 .

[30]  I. Singer,et al.  Surrogate duality for vector optimization , 1987 .

[31]  P. Pardalos,et al.  Optimization and optimal control , 2003 .

[32]  Clarence Zener,et al.  Geometric Programming , 1974 .

[33]  I. Singer,et al.  An extension of D.C. duality theory, with an appendix on ∗-subdifferentials , 1997 .

[34]  Alexander Rubinov,et al.  Lipschitz programming via increasing convex-along-rays functions * , 1999 .

[35]  Pham Dinh Tao,et al.  Duality in D.C. (Difference of Convex functions) Optimization. Subgradient Methods , 1988 .

[36]  L. McKenzie,et al.  Demand Theory Without a Utility Index , 1957 .

[37]  Phan Thien Thach A Nonconvex Duality with Zero Gap and Applications , 1994, SIAM J. Optim..

[38]  I. Singer,et al.  Generalized Convexity, Functional Hulls and Applications to Conjugate Duality in Optimization , 1984 .

[39]  C. Scott,et al.  Conjugate duality in generalized fractional programming , 1989 .

[40]  J. Henry,et al.  System Modelling and Optimization: Proceedings of the 16th IFIP-TC7 Conference, Compiègne, France, July 5-9, 1993 , 1994, System Modelling and Optimization.

[41]  Michel Volle QUASICONVEX DUALITY FOR THE MAX OF TWO FUNCTIONS , 1997 .

[42]  B. Simsek,et al.  Conjugate quasiconvex nonnegative functions , 1995 .

[43]  Juan Enrique Martínez-Legaz,et al.  Dualities Associated To Binary Operations On R , 1995 .

[44]  Yair Censor,et al.  Recent Developments in Optimization Theory and Nonlinear Analysis: Ams/Imu Special Session on Optimization and Nonlinear Analysis, May 24-26, 1995, Jerusalem, Israel , 1997 .

[45]  Jacques A. Ferland,et al.  Duality in generalized linear fractional programming , 1984, Math. Program..

[46]  E. Barron,et al.  Hope-lax type formular for u∞ t +H u,D u =0: II , 1997 .

[47]  I. Singer,et al.  A Fenchel-Rockafellar type duality theorem for maximization , 1979, Bulletin of the Australian Mathematical Society.

[48]  Dual problems of quasiconvex maximisation , 1995, Bulletin of the Australian Mathematical Society.

[49]  E. N. Barron,et al.  Applications of the Hopf-Lax formula for u t + H ( u,Du ) = 0 , 1998 .

[50]  J. Hiriart-Urruty,et al.  Trends in Mathematical Optimization , 1987 .

[51]  Masakazu Kojima,et al.  A Generalized Convexity and Variational Inequality for Quasi-Convex Minimization , 1996, SIAM J. Optim..

[52]  B. M. Glover,et al.  Cutting angle methods in global optimization , 1999 .

[53]  Juan Enrique Martínez-Legaz,et al.  On lower subdifferentiable functions , 1988 .

[54]  Juan Enrique Martínez-Legaz,et al.  Duality in D.C. Programming: The Case of Several D.C. Constraints , 1999 .

[55]  A. Elqortobi Conjugaison quasi convexe des fonctionnelles positives , 1993 .

[56]  W. Ziemba,et al.  Generalized concavity in optimization and economics , 1981 .

[57]  Juan Enrique Martínez-Legaz,et al.  On Expenditure Functions , 1996 .

[58]  J. Martínez-Legaz Axiomatic characterizations of the duality correspondence in consumer theory , 1993 .

[59]  P. Hammond,et al.  Handbook of Utility Theory , 2004 .

[60]  Dual approaches to utility , 1993 .

[61]  Ivan Singer Some relations between dualities, polarities, coupling functionals, and conjugations , 1986 .

[62]  H. Ishii,et al.  Hopf-Lax Formulas for Semicontinuous Data , 1999 .

[63]  A. M. Rubinov,et al.  Quasiconvexity via Two Step Functions , 1998 .

[64]  J. P. Penot DUALITY FOR RADIANT AND SHADY PROGRAMS , 1997 .

[65]  Richard Cornes,et al.  Duality and modern economics , 1993 .

[66]  A. Rubinov Abstract Convexity and Global Optimization , 2000 .

[67]  I. Singer Duality in quasi-convex supremization and reverse convex infimization via abstract convex analysis,and applications to approximation ** , 1999 .

[68]  Jean-Paul Penot,et al.  Duality for Anticonvex Programs , 2001, J. Glob. Optim..

[69]  F. Glover A Multiphase-Dual Algorithm for the Zero-One Integer Programming Problem , 1965 .

[71]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[72]  R. N. Mukherjee Generalized convex duality for multiobjective fractional programs , 1991 .

[73]  M. Volle Contributions à la dualité en optimisation et à l'épi-convergence , 1986 .

[74]  Bernard Lemaire,et al.  A General Duality Scheme for Nonconvex Minimization Problems with a Strict Inequality Constraint , 1998, J. Glob. Optim..

[75]  R. Shepherd Theory of cost and production functions , 1970 .

[76]  Ivan Singer,et al.  On Conjugations for Functions with Values in Extensions of Ordered Groups , 1997 .

[77]  Diethard Pallaschke,et al.  Foundations of Mathematical Optimization , 1997 .

[78]  E. N. Barron,et al.  Calculus of variations in L , 1997 .

[79]  Duality and sufficiency for fractional control problems with invexity , 1999 .

[80]  Michael E. Thomas,et al.  Mathematical methods in operations research , 1968 .

[81]  Diethard Pallaschke,et al.  Selected Topics in Operations Research and Mathematical Economics , 1984 .

[82]  J. E. Martínez-Legaz Characterization of R-Evenly Quasiconvex Functions , 1997 .

[83]  Phan Thien Thach,et al.  Diewert-Crouzeix conjugation for general quasiconvex duality and applications , 1995 .

[84]  David G. Luenberger,et al.  Quasi-Convex Programming , 1968 .

[85]  A note on duality in consumer theory , 1999 .

[86]  S. Schaible,et al.  4. Application of Generalized Concavity to Economics , 1988 .

[87]  Juan Enrique Martínez-Legaz,et al.  V -dualities and ⊥-dualities , 1991 .

[88]  H. Houthakker Revealed Preference and the Utility Function , 1950 .

[89]  Rita Pini,et al.  A survey of recent[1985-1995]advances in generalized convexity with applications to duality theory and optimality conditions , 1997 .

[90]  W. Gangbo,et al.  The geometry of optimal transportation , 1996 .

[91]  Semen S. Kutateladze,et al.  MINKOWSKI DUALITY AND ITS APPLICATIONS , 1972 .

[92]  P. Samuelson A Note on the Pure Theory of Consumer's Behaviour , 1938 .

[93]  Juan Enrique Martínez-Legaz Lexicographical order, inequality systems and optimization , 1984 .

[94]  Juan Enrique Martínez-Legaz,et al.  Dualities between complete lattices , 1990 .

[95]  Juan Enrique Martínez-Legaz,et al.  Subdifferentials with respect to dualities , 1995, Math. Methods Oper. Res..

[96]  M. Volle Conjugaison par tranches et dualitë de toland , 1987 .

[97]  M. Atteia,et al.  Quasi-convex duality , 1981 .

[98]  Michel Volle Conjugaison par tranches , 1985 .

[99]  Ivan Singer,et al.  A general theory of dual optimization problems. II: On the perturbational dual problem corresponding to an unperturbational dual problem , 1989, ZOR Methods Model. Oper. Res..

[100]  M. Volle A Formula on the Conjugate of the Max of a Convex Function and a Concave Function , 1998 .

[101]  Juan Enrique Martínez-Legaz,et al.  α-Lower Subdifferentiable Functions , 1993, SIAM J. Optim..

[102]  J. Martínez-Legaz,et al.  A contribution to duality theory, applied to the measurement of risk aversion , 2006 .

[103]  Werner Oettli,et al.  Conjugate Functions for Convex and Nonconvex Duality , 1998, J. Glob. Optim..

[105]  A. M. Rubinov,et al.  Increasing Convex-Along-Rays Functions with Applications to Global Optimization , 1999 .

[106]  R. Rockafellar Conjugate Duality and Optimization , 1987 .

[107]  J. Toland Duality in nonconvex optimization , 1978 .

[108]  Juan Enrique Martínez-Legaz,et al.  Quasiconvex duality theory by generalized conjugation methods , 1988 .

[109]  Phan Thien Thach Quasiconjugates of functions, duality relationship between Quasiconvex Minimization under a Reverse Convex Constraint and Quasiconvex Maximization under a Convex Constraint, and applications☆ , 1991 .

[110]  Michel Volle,et al.  Concave duality: Application to problems dealing with difference of functions , 1988, Math. Program..

[111]  J. Penot What is quasiconvex analysis? , 2000 .

[112]  Quasi-convex Functions and Quasi-monotone Operators , 1995 .

[113]  Juan Enrique Martínez-Legaz Exact quasiconvex conjugation , 1983, Z. Oper. Research.

[114]  Alfred Auslender,et al.  Convex Analysis and Its Applications , 1977 .

[115]  Adil M. Bagirov,et al.  Global Minimization of Increasing Positively Homogeneous Functions over the Unit Simplex , 2000, Ann. Oper. Res..

[116]  I. Singer Infimal generators and dualities between complete lattices , 1987 .

[117]  W. A. Thompson,et al.  Some Properties of Generalized Concave Functions , 1973, Oper. Res..

[118]  Szymon Dolecki,et al.  On $\Phi $-Convexity in Extremal Problems , 1978 .

[119]  Jean-Paul Penot,et al.  Another duality scheme for Quasiconvex Problems , 1988 .

[120]  Erik J. Balder,et al.  An Extension of Duality-Stability Relations to Nonconvex Optimization Problems , 1977 .

[121]  M. Richter Duality and rationality , 1979 .

[122]  Jean-Paul Penot,et al.  Convexity and Generalized Convexity Methods for the Study of Hamilton-Jacobi Equations , 2001 .

[123]  M. Volle Duality for the level sum of quasiconvex functions and applications , 1998 .

[124]  Atemporal, Temporal and Intertemporal Duality in Consumer Theory , 1980 .

[125]  P. Samuelson A Note on the Pure Theory of Consumer's Behaviour: An Addendum , 1938 .

[126]  Nicolas Hadjisavvas,et al.  On the Subdifferentials of Quasiconvex and Pseudoconvex Functions and Cyclic Monotonicity , 1999 .

[127]  Yakar Kannai A characterization of monotone individual demand functions , 1989 .

[128]  Jean-Paul Penot,et al.  On Quasi-Convex Duality , 1990, Math. Oper. Res..

[129]  Ivan Singer A general theory of dual optimization problems , 1986 .

[130]  J. P. Crouzeix Conjugacy in quasiconvex analysis , 1977 .

[131]  Ilya Segal,et al.  Solutions manual for Microeconomic theory : Mas-Colell, Whinston and Green , 1997 .

[132]  Equations inf-convolutives et conjugaison de Moreau Fenchel , 1991 .

[133]  Siegfried Schaible,et al.  Increasing quasiconcave co-radiant functions with applications in mathematical economics , 2005, Math. Methods Oper. Res..

[134]  V. Levin Abstract Cyclical Monotonicity and Monge Solutions for the General Monge–Kantorovich Problem , 1999 .

[135]  Jean-Paul Penot,et al.  Generalized Convexity and Generalized Monotonicity , 2001 .

[136]  D. Bridges,et al.  Representations of Preferences Orderings , 1995 .

[137]  Harvey J. Greenberg,et al.  Surrogate Mathematical Programming , 1970, Oper. Res..

[138]  Diethard Pallaschke,et al.  Foundations of mathematical optimization : convex analysis without linearity , 1997 .

[139]  J. Quah The Monotonicity of Individual and Market Demand , 2000 .

[140]  Juan Enrique Martínez-Legaz,et al.  Characterizations of evenly convex sets and evenly quasiconvex functions , 2002 .

[141]  Carlton H. Scott,et al.  Duality of a Nonconvex Sum of Ratios , 1998 .

[142]  R. Rockafellar Augmented Lagrange Multiplier Functions and Duality in Nonconvex Programming , 1974 .

[143]  Juan Enrique Martínez-Legaz,et al.  Dual representation of cooperative games based on fenchel-moreau conjugation , 1996 .