Division of the Humanities
and Social Sciences

Ec 181: Convex Analysis and Economic Theory

Winter 2020

Time: Tuesday, Thusday, 2:30–4:55 pm // Location: 127 Baxter Hall

Professor: KC Border
Email: kcb@caltech.edu

Woe to the author who always wants to teach!
The secret of being a bore is to tell everything.

—Voltaire, De la Nature de l'Homme (1737)

This is an evolving course that is designed to introduce you to convex analysis and its applications in economics. It is still under development.

Convex analysis is the study of the properties of convex sets and convex and concave functions. The fundamental results in the field are the separating hyperplane theorems. These results have interpretations as existence theorems for prices, so they are fundamental in many areas of economic theory.

Another class of theorems goes by the name of the Theorem of the Alternative. These theorems give conditions for the existence of solutions to linear inequalities in terms of the existence of solutions to an alternative set of inequalities. This may not seem especially useful, but these results are at the heart of fundamental results in decision theory and asset pricing theory.

The goal of this course is to present the useful results from convex analysis in a way that you understand their proofs and can use them in economics. There will be a lot of proofs in this course, and you will be expected to prove things. If you do not like proving theorems, you should not take this course.

The course will concentrate on convex analysis in finite dimensional spaces, but I will also discuss infinite dimensional spaces (which are necessary in mathematical finance) whenever it seems productive to do so. In particular, I will try to avoid making use of the dimensionality of the space whenever possible. But some things that are true for finite dimensional spaces are not true for infinite dimensional spaces.

Evaluation

This is a seminar course. Students are expected to show up and participate. Students will be responsible for presenting the bulk of the material, incuding preparation of notes for the lecture. There may be a few exercises assigned, and students may be selected at random to explain the exercises. Here is a partial list of optional topics (2011-09-27, 10:08).

The grade will be based on participation (40%) and presentations (60%). Presentations will be evaluated on both the written notes and the lecture.

Exercises

Text

There is no required textbook for the class. I will make my own notes available via the web. I expect these to change over the course of the term based on feedback that I receive from you. For those of you who like having a textbook, I recommend Optima and Equilibria: An Introduction to Nonlinear Analysis by J.-P. Aubin, Springer-Verlag, 1993; Fundamentals of Convex Analysis by J.-B. Hiriart-Urruty and C. Lemaréchal, Springer--Verlag, 2001; and Convex Analysis and Nonlinear Optimization: Theory and Examples by J. M. Borwein and A. S. Lewis, Springer, 2006.

Notes on Topics

These notes are only a guide to the fundamental material, and will be revised over the course of the term.

Table of Contents (2020-03-10, 13:18)
Topic 0 (2019-12-23, 02:49)
Topic 1 (2019-12-23, 02:49)
Topic 2 (2019-12-23, 02:49)
Topic 3 (2019-11-18, 14:06)
Topic 4 (2019-12-23, 02:49)
Topic 5 (2019-12-23, 02:49)
Topic 6 (2019-12-23, 02:49)
Topic 7 (2019-12-23, 02:49)
Topic 8 (2019-12-23, 02:49)
Topic 9 (2019-12-22, 19:00)
Topic 10 (2019-12-23, 02:49)
Topic 11 (2019-12-23, 02:49)
Topic 12 (2019-12-24, 12:34)
Topic 13 (2019-12-23, 02:50)
Topic 14 (2019-12-23, 02:50)
Topic 15 (2019-11-18, 14:08)
Topic 16 (2019-12-23, 02:50)
Topic 17 (2019-12-24, 12:34)
Topic 18 (2019-12-22, 19:00)
Topic 19 (2019-11-18, 14:08)
Topic 20 (2019-11-18, 14:08)
Topic 21 (2020-03-10, 13:18)
Topic 22 (2019-12-24, 12:35)
Topic 23 (2019-12-24, 12:35)
Topic 24 (2019-12-24, 12:35)
Topic 25 (2020-03-10, 13:18)
Topic 26 (2019-12-23, 02:50)
Topic 27 (2020-03-10, 13:18)
Topic 28 (2020-01- 7, 13:50)
Topic 29 (2020-01- 9, 16:43)
Appendix (2019-12-22, 19:01)
What you should remember about metric spaces. (2018-10-15, 15:30)
A quick introduction to the concepts of point-set topology. (2018-10- 3, 13:35)
A quick introduction to correspondences. (2016-09-11, 01:15)

Tentative outline

Here is a tentative course outline. I'm not sure how fast we can cover this material, so expect it to change over the course of the term.

Overview of some of the applications
- Cost and production functions. Shephard's Lemma and Hotelling's Lemma.
- Second welfare theorem.
- Core convergence in replica economies
- Applications in decision theory. Results of de Finnetti, Scott, Chambers, Border, and Ledyard.
- Linear programming and its application to zero-sum two-person games, the core of a TU game, and incentive design.
- Asset pricing
Review of linear algebra. Hilbert spaces. Topological vector spaces.
Separating hyperplane theorems.
Support points and supporting hyperplanes.
Second welfare theorem and core convergence in replica economies
Support functionals and duality
Affine functions and hyperplanes
Convex and Concave functions.
Semicontinuity, closedness, and majorization
Subgradients and subdifferentials
Continuity and Differentiability of convex functions
Cost and production functions. Shephard's Lemma and Hotelling's Lemma.
Cyclical monotonicity
Fenchel conjugacy
Calculus of Subdifferentials
Theorem of the Alternative (geometric approach)
Theorem of the Alternative (algebraic approach)
Applications in decision theory. Results of de Finnetti, Scott, Chambers, Border, and Ledyard.
Linear programming and its application to zero-sum two-person games, the core of a TU game, and incentive design.
Asset pricing
Recession cones and closedness of sets
Finite cones and polyhedral convexity.
Fourier–Motzkin Elimination

Additional Readings and Historical Articles

Convex Analysis

Ilan Adler, Nimrod Megiddo, and Michael J. Todd. 1984. New results on the average behavior of simplex algorithms. Bulletin of the American Mathematical Society 11: 378–382. On-line at Project Euclid
A. D. Alexandroff. 1939. Almost Everywhere Existence of the Second Differential of a Convex Function and Surfaces Connected with it. Leningrad State University Annals, Mathematics Series 6: 3–35. In Russian.
Charalambos D. Aliprantis and KC Border. 2006. Infinite Dimensional Analysis: A Hitchhiker's Guide. Springer–Verlag, Berlin.
E. J. Anderson and P. Nash. 1987. Linear Programming in Infinite Dimensional Spaces. John Wiley and Sons, New York.
H. A. Antosiewicz, ed. 1955. Proceedings of the Second Symposium in Linear Programming. National Bureau of Standards and Directorate of Management Analysis, DCS/Comptroller, USAF, Washington, D.C.
Aloisio Araujo. 1991. The Once but Not Twice Differentiability of the Policy Function. Econometrica 59: 1383–1393. On-line at JSTOR
V. I. Arkin and V. L. Levin. 1971. Extreme Points of a Certain Set of Measurable Vector Functions of Several Variables and Convexity of the Values of Vector Integrals. Soviet Mathematics–Doklady 12: 1248–1252.
Edgar Asplund and R. T. Rockafellar. 1969. Gradients of Convex Functions. Transactions of the American Mathematical Society 139: 443–467. On-line at JSTOR
Achim Bachem and Walter Kern. 1992. Linear Programming Duality: An Introduction to Oriented Matroids. Springer–Verlag, Berlin. ISBN: 978-3-540-55417-2.
M. L. Balinski. 1961. An Algorithm for Finding All Vertices of Convex Polyhedral Sets. Journal of the Society for Industrial and Applied Mathematics 9: 72–88. On-line at JSTOR
Gerald A. Beer. 1993. Topologies on Closed and Closed Convex Sets. Kluwer Academic Publishers, Dordrecht.
Gerald A. Beer. 1994. Lipschitz Regularization and the Convergence of Convex Functions. Numerical Functional Analysis and Optimization 15: 31–46.
A. Ben-Israel and B. Mond. 1986. What is Invexity? Journal of the Australian Mathematical Society, Series B 28: 1–9. DOI: 10.1017/S0334270000005142
Joël Benoist, Jonathan M. Borwein, and Nicolae Popovici. 2003. A Characterization of Quasiconvex Vector-Valued Functions. Proceedings of the American Mathematical Society 131: 1109–1113. On-line at JSTOR
Abraham Berman and Adi Ben-Israel. 1971. Linear inequalities, mathematical programming and matrix theory. Mathematical Programming 1: 291–300. DOI: 10.1007/BF01584093
Dimitri P. Bertsekas. 2002. Convexity and Optimization. M.I.T. Lecture Notes. On-line at MIT
Dimitri P. Bertsekas, Angelia Nedić, and Asuman E. Ozdaglar. 2002. Min Common/Max Crossing Duality: A Simple Geometric Framework for Convex Optimization and Minimax Theory. Dept. of Electrical Engineering and Computer Science, Laboratory for Information and Decision Systems, M.I.T. LIDS-P no. 2536.
Dimitri P. Bertsekas and Asuman E. Ozdaglar. 2000. Pseudonormality and a Lagrange Multiplier Theory for Constrained Optimization. Manuscript.
Errett Bishop and Robert R. Phelps. The Support Functionals of a Convex Set. In Klee1963.
KC Border. 1985. Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press, New York.
Jonathan M. Borwein and Chris H. Hamilton. 2009. Symbolic Fenchel Conjugation. Mathematical Programming, Series B 116: 17–35. DOI: 10.1007/s10107-007-0134-4
J. M. Borwein and A. S. Lewis. 1991. Duality Relationships for Entropy-Like Minimization Problems. SIAM Journal on Control and Optimization 29: 325-338. DOI: 10.1137/0329017
Jonathan M. Borwein and Adrian S. Lewis. 2006. Convex Analysis and Nonlinear Optimization: Theory and Examples. Springer, New York.
J. M. Borwein and David Preiss. 1987. A Smooth Variational Principle with Applications to Subdifferentiability and to Differentiability of Convex Functions. Transactions of the American Mathematical Society 303: 517–527. On-line at JSTOR
Jonathan M. Borwein and Qiji J. Zhu. 1999. A Survey of Subdifferential Calculus with Applications. Nonlinear Analysis 38: 687–773. DOI: 10.1016/S0362-546X(98)00142-4
Jonathan M. Borwein and Qiji J. Zhu. 2005. Techniques of Variational Analysis. Springer Science+Business Media, New York.
Jonathan M. Borwein and Qiji J. Zhu. 2006. Variational Methods in Convex Analysis. Journal of Global Optimization 35: 197–213. DOI: 10.1007/s10898-005-3835-3
G. Bouchitté. 2006. Convex Analysis and Duality Methods. In Encyclopedia of Mathematical Physics. Jean-Pierre Françoise, Gregory L. Naber, and Tsou S. Tsun, ed. Academic Press, Oxford. DOI: 10.1016/B0-12-512666-2/00487-9
Stephen Boyd and Lieven Vandenberghe. 2004. Convex Optimization. Cambridge University Press, Cambridge and New York. ISBN: 9780521833783.
C. C. Braunschweiger. 1962. An Extension of the Nonhomogeneous Farkas Theorem. American Mathematical Monthly 69: 969–975. On-line at JSTOR
C. C. Braunschweiger and H. E. Clark. 1962. An Extension of the Farkas Theorem. American Mathematical Monthly 69: 272–277. On-line at JSTOR
A. Brøndsted and R. T. Rockafellar. 1965. On the Subdifferentiability of Convex Functions. Proceedings of the American Mathematical Society 16: 605–611. On-line at JSTOR
Felix E. Browder. 1965. Nonlinear Monotone Operators and Convex Sets in Banach Spaces. Bulletin of the American Mathematical Society 71: 780-785. On-line at Project Euclid
Charles Castaing and Michel Valadier. 1977. Convex Analysis and Measurable Multifunctions. Springer–Verlag, Berlin.
Yves Chabrillac and J. -P. Crouzeix. 1984. Definiteness and semidefiniteness of quadratic forms revisited. Linear Algebra and its Applications 63: 283–292. DOI: 10.1016/0024-3795(84)90150-2
Thomas Christof and Andreas Loebel. 1997–2002. PORTA—A Polyhedron Representation Transformation Algorithm. On-line Version 1.4.0; source code available from the University of Heidelberg.
Frank H. Clarke. 1976. A New Approach to Lagrange Multipliers. Mathematics of Operations Research 1: 165–174. On-line at JSTOR
Jean-Pierre Crouzeix and Jacques A. Ferland. 1982. Criteria for quasi-convexity and pseudo-convexity: Relationships and comparisons. Mathematical Programming 23: 193-205. DOI: 10.1007/BF01583788
George B. Dantzig. A Proof of the Equivalence of the Programming Problem and the Game Problem. In Koopmans1951. On-line at the Cowles Commission
George B. Dantzig. 1960. On the Significance of Solving Linear Programming Problems with Some Integer Variables. Econometrica 28: 30–44.
George B. Dantzig. 1963. Linear Programming and Extensions. Princeton University Press, Princeton.
George B. Dantzig and B. C. Eaves. 1973. Fourier–Motzkin Elimination and its Dual. Journal of Combinatorial Theory, Series A 14: 288–297. DOI: 10.1016/0097-3165(73)90004-6
George B. Dantzig, Alex Orden, and Philip Wolfe. 1954. Notes on Linear Programming: Part I—the Generalized Simplex Method for Minimizing a Linear Form under Linear Inequality Constraints. The RAND Corporation. Research Memorandum no. RM-1264. April 5, 1954. 17pp.
George B. Dantzig, Alex Orden, and Philip Wolfe. 1955. The Generalized Simplex Method for Minimizing a Linear Form under Linear Inequality Constraints. Pacific Journal of Mathematics 5: 183–195. On-line at Project Euclid
George B. Dantzig and Mukund N. Thapa. 2003. Linear Programming: 2. Theory and Extensions. Springer–Verlag, New York.
George B. Dantzig and Marshall K. Wood. 1949. Programming of Interdependent Activities: I General Discussion. Econometrica 17: 193–1999. On-line at JSTOR
Gerard Debreu. 1964. Nonnegative Solutions of Linear Inequalities. International Economic Review 5: 178–184. On-line at JSTOR
Bruno de Finetti. 1949. Sulle Stratificazioni Convesse. Annali di Matematica Pura ed Applicata. Serie 4 30: 173–183. DOI: 10.1007/BF02415006
Erwin E. Diewert, M. Avriel, and I. Zang. 1981. Nine kinds of quasiconcavity and concavity. Journal of Economic Theory 25: 397–420. DOI: 10.1016/0022-0531(81)90039-9
Lloyd L. Dines. 1919. Systems of Linear Inequalities. Annals of Mathematics 20: 191–199. On-line at JSTOR
Lloyd L. Dines. 1926. On Positive Solutions of a System of Linear Equations. Annals of Mathematics 28: 386–392. On-line at JSTOR
Lloyd L. Dines. 1938. On Convexity. American Mathematical Monthly 45: 199–209. On-line at JSTOR
R. J. Duffin and L. A. Karlovitz. 1965. An Infinite Linear Program with a Duality Gap. Management Science 12: 122–134. On-line at JSTOR
Geir E. Dullerud and Fernando Paganini. 2000. A Course in Robust Control Theory: A Convex Approach. Springer–Verlag, New York.
H. G. Eggleston. 1958. Convexity. Cambridge University Press, Cambridge.
E. Eisenberg. 1961. Duality in Homogeneous Programming. Proceedings of the American Mathematical Society 12: 783–787. On-line at JSTOR
Ivar Ekeland and Roger Temam. 1976. Convex Analysis and Variational Problems. North Holland, Amsterdam.
Ivar Ekeland and Thomas Turnbull. 1983. Infinite-Dimensional Optimization and Convexity. University of Chicago Press, Chicago.
Ky Fan. 1952. A Generalization of Tucker's Combinatorial Lemma with Topological Applications. Annals of Mathematics 56: 431–437. On-line at JSTOR
Ky Fan. On systems of linear inequalities. In KuhnTucker1956.
Ky Fan. 1964. Sur Une Théorème Minimax. Comptes Rendus des Séances de l'Académie des Sciences (Paris), Groupe 1 259: 3925–3928.
Ky Fan. 1969. Asymptotic cones and duality of linear relations. Journal of Approximation Theory 2: 152–159. DOI: 10.1016/0021-9045(69)90038-0
Ky Fan. 1969. Extensions of Two Fixed Point Theorems of F. E. Browder. Mathematische Zeitschrift 112: 234–240.
Ky Fan. 1972. A Minimax Inequality and Applications. In Inequalities III. O. Shisha, ed. Academic Press, New York.
Ky Fan. 1984. Some Properties of Convex Sets Related to Fixed Point Theorems. Mathematische Annalen 266: 519–537. DOI: 10.1007/BF01458545
Ky Fan, Irving Glicksberg, and A. J. Hoffman. 1957. Systems of Inequalities Involving Convex Functions. Proceedings of the American Mathematical Society 8: 617–622. On-line at JSTOR
Julius Farkas. 1902. Über Die Theorie Der Einfachen Ungleichungen. Journal für die Reine und Angewandte Mathematik 124: 1–27. On-line at DigiZeitschriften
Werner Fenchel. 1949. On conjugate convex functions. Canadian Journal of Mathematics 1: 73–77. On-line at the Canadian Mathematical Society
Werner Fenchel. 1953. Convex Cones, Sets, and Functions. Princeton University, Department of Mathematics. Lecture Notes. From notes taken by D. W. Blackett, Spring 1951.
Jacques A. Ferland. 1981. Matrix-theoretic criteria for the quasiconvexity of twice continuously differentiable functions. Linear Algebra and its Applications 38: 51–63. DOI: 10.1016/0024-3795(81)90007-0
Peter C. Fishburn. 1988. Combinatorial Optimization Problems for Systems of Subsets. SIAM Review 30: 578–588. On-line at JSTOR
Monique Florenzano and Cuong Le Van. 2001. Finite Dimensional Convexity and Optimization. Springer–Verlag, New York and Heidelberg.
Joel Franklin. 1980. Methods of Mathematical Economics. Springer–Verlag, New York.
Joel Franklin. 2002. Methods of Mathematical Economics: Linear and Nonlinear Programming, Fixed Point Theorems. SIAM, Philadelphia. Corrected reprint of the 1980 edition published by Springer–Verlag.
Komei Fukuda. 2004. Frequently Asked Questions in Polyhedral Computation. Swiss Federal Institute of Technology. On-line
Komei Fukuda and Alain Prodon. 1996. Double Description Method Revisited. In Combinatorics and Computer Science. M. Deza, R. Euler, and I. Manoussakis, ed. Springer–Verlag, Berlin. On-line
Delbert R. Fulkerson. 1966. Flow Networks and Combinatorial Operations Research. American Mathematical Monthly 73: 115–138. On-line at JSTOR
Jerry W. Gaddum. 1952. A Theorem on Convex Cones with Applications to Linear Inequalities. Proceedings of the American Mathematical Society 3: 957–960. On-line at JSTOR
David Gale. Convex Polyhedral Cones and Linear Inequalities. In Koopmans1951. On-line at the Cowles Commission
David Gale. 1964. On the number of faces of a convex polytope. Canadian Journal of Mathematics 16: 12–17. DOI: 10.4153/CJM-1964-002-x
David Gale. 1989. Theory of Linear Economic Models. University of Chicago Press, Chicago. Reprint of the 1960 edition published by McGraw-Hill.
David Gale. 1969. How to Solve Linear Inequalities. American Mathematical Monthly 76: 589–599. On-line at JSTOR
David Gale, Victor Klee, and R. T. Rockafellar. 1968. Convex Functions on Convex Polytopes. Proceedings of the American Mathematical Society 19: 867–873. On-line at JSTOR
David Gale, Harold W. Kuhn, and Albert W. Tucker. Linear Programming and the Theory of Games. In Koopmans1951. On-line at the Cowles Commission
David Gale, Harold W. Kuhn, and Albert W. Tucker. Linear Programming and the Theory of Games. In Koopmans1951. On-line at the Cowles Commission
Murray Gerstenhaber. Theory of Convex Polyhedral Cones. In Koopmans1951. On-line at the Cowles Commission
John R. Giles. 1982. Convex Analysis with Application in Differentiation of Convex Functions. Pitman Advanced Publishing Program, Boston.
A. J. Goldman and Albert W. Tucker. Polyhedral Convex Cones. In KuhnTucker1956.
P. Gordan. 1873. Über Die Auflösung Linearer Gleichungen Mit Reelen Coefficienten [On the Solution of Linear Inequalities with Real Coefficients]. Mathematische Annalen 6: 23–28. DOI: 10.1007/BF01442864
Neil E. Gretsky, Joseph M. Ostroy, and William R. Zame. 2002. Subdifferentiability and the Duality Gap. Positivity 6: 261–274. On-line
Hubert Halkin. 1966. Necessary and Sufficient Condition for a Convex Set to be Closed. American Mathematical Monthly 73: 628–630. On-line at JSTOR
Hubert Halkin. 1966. A Property of Nonseparated Convex Sets. Proceedings of the American Mathematical Society 17: 1389–1395. On-line at JSTOR
G. H. Hardy, J. E. Littlewood, and G. Pólya. 1929. Some Simple Inequalities Satisfied by Convex Functions. Messenger of Mathematics 58: 145–152.
Jean-Baptiste Hiriart-Urruty and Claude Lemaréchal. 1993. Convex Analysis and Minimization Algorithms I. Springer–Verlag, Berlin.
Jean-Baptiste Hiriart-Urruty and Claude Lemaréchal. 1993. Convex Analysis and Minimization Algorithms II. Springer–Verlag, Berlin.
Jean-Baptiste Hiriart-Urruty and Claude Lemaréchal. 2001. Fundamentals of Convex Analysis. Springer–Verlag, Berlin.
Lars Hörmander. 1954. Sur La Fonction D'Appui Des Ensembles Convexes Dans Un Espace Localement Convexe. Arkiv för Matematik 3: 181–186.
Hendrik S. Houthakker. 1960. The Capacity Method of Quadratic Programming. Econometrica 28: 62–87. On-line at JSTOR
Hendrik S. Houthakker. 1962. The Capacity Method of Quadratic Programming (Errata). Econometrica 30: 633. On-line at JSTOR
Ralph Howard. 1998. Alexandrov's Theorem on the Second Derivatives of Convex Functions Via Rademacher's Theorem on the First Derivatives of Lipschitz Functions. Department of Mathematics, University of South Carolina. Lecture notes. On-line
John E. Jayne and C. A. Rogers. 1977. The Extremal Structure of Convex Sets. Journal of Functional Analysis 26: 251–288.
Samuel Karlin. 1959. Mathematical Methods and Theory in Games, Programming, and Economics. Addison-Wesley, Reading, MA.
Samuel Karlin. 1987. Mathematical Methods and Theory in Games, Programming, and Economics. Dover, New York. Reprint of the 1959 two-volume edition published by Addison–Wesley.
Samuel Karlin and A. Novikoff. 1963. Generalized Convex Inequalities. Pacific Journal of Mathematics 13: 1251–1279. On-line at Project Euclid
Victor L. Klee, Jr. 1948. The Support Property of a Convex Set. Duke Mathematical Journal 15: 767–772. On-line at Project Euclid
Victor L. Klee, Jr. 1949. A Characterization of Convex Sets. American Mathematical Monthly 56: 247–249. On-line at JSTOR
Victor L. Klee, Jr. 1950. Decomposition of an Infinite-Dimensional Linear System into Ubiquitous Convex Sets. American Mathematical Monthly 57: 540–541. On-line at JSTOR
Victor L. Klee, Jr. 1951. Some Characterizations of Compactness. American Mathematical Monthly 58: 389–393. On-line at JSTOR
Victor L. Klee, Jr. 1951. Convex Sets in Linear Spaces. Duke Mathematical Journal 18: 443–466. DOI: 10.1215/S0012-7094-51-01835-2
Victor L. Klee, Jr. 1951. Convex Sets in Linear Spaces, II. Duke Mathematical Journal 18: 875–883. DOI: 10.1215/S0012-7094-51-01882-0
Victor L. Klee, Jr. 1953. Convex Sets in Linear Spaces, III. Duke Mathematical Journal 20: 105–111. DOI: 10.1215/S0012-7094-53-02010-9
Victor L. Klee, Jr. 1953. Convex Bodies and Periodic Homeomorphisms in Hilbert Space. Transactions of the American Mathematical Society 74: 10–43. On-line at JSTOR
Victor L. Klee, Jr. 1955. A Note on Extreme Points. American Mathematical Monthly 62: 30–32. On-line at JSTOR
Victor L. Klee, Jr. 1955. Separation Properties of Convex Cones. Proceedings of the American Mathematical Society 6: 313–318. On-line at JSTOR
Victor L. Klee, Jr. 1955. Some Topological Properties of Convex Sets. Transactions of the American Mathematical Society 78: 30–45. On-line at JSTOR
Victor L. Klee, Jr. 1956. Strict Separation of Convex Sets. Proceedings of the American Mathematical Society 7: 735–737. On-line at JSTOR
Victor L. Klee, Jr. 1963. Convexity. American Mathematical Society, Providence, RI.
Victor L. Klee, Jr. 1963. On a Question of Bishop and Phelps. American Journal of Mathematics 85: 95–98. On-line at JSTOR
Victor L. Klee, Jr. 1964. On the number of vertices of a convex polytope. Canadian Journal of Mathematics 16: 701–720. DOI: 10.4153/CJM-1964-067-6
Victor L. Klee, Jr. 1971. What is a Convex Set? American Mathematical Monthly 78: 616–631. On-line at JSTOR
Tjalling C. Koopmans, ed. 1951. Activity Analysis of Production and Allocation: Proceedings of a Conference. John Wiley and Sons, New York. On-line at the Cowles Commission
M. G. Krein and D. Milman. 1940. On Extreme Points of Regular Convex Sets. Studia Mathematica 9: 133–138. On-line at the Polish Virtual Library of Science
M. G. Krein and V. L. Šmulian. 1940. On Regularly Convex Sets in the Space Conjugate to a Banach Space. Annals of Mathematics 41: 556–583. On-line at JSTOR
Harold W. Kuhn. 1956. Solvability and Consistency for Linear Equations and Inequalities. American Mathematical Monthly 63: 217–232. On-line at JSTOR
Harold W. Kuhn and Albert W. Tucker, ed. 1950. Contributions to the Theory of Games, I. Princeton University Press, Princeton.
Harold W. Kuhn and Albert W. Tucker, ed. 1953. Contributions to the Theory of Games, II. Princeton University Press, Princeton.
Harold W. Kuhn and Albert W. Tucker, ed. 1956. Linear Inequalities and Related Systems. Princeton University Press, Princeton.
R. Lehmann and W. Oettli. 1975. The theorem of the alternative, the key-theorem, and the vector-maximum problem. Mathematical Programming 8: 332–344. DOI: 10.1007/BF01580450
Alain Leroux. 1984. Other determinantal conditions for concavity and quasi-concavity. Journal of Mathematical Economics 13: 43–49. DOI: 10.1016/0304-4068(84)90023-5
Marco Longinetti and Paolo Salani. 2007. On the Hessian matrix and Minkowski addition of quasiconvex functions. Journal de Mathématiques Pures et Appliquées. Neuvième Série 88: 276 - 292. DOI: 10.1016/j.matpur.2007.06.007
Dinh T. Luc. 1986. Random Version of the Theorems of the Alternative. Mathematische Nachrichten 129: 149–155. DOI: 10.1002/mana.19861290113
Yves Lucet. 2010. What Shape is Your Conjugate? A Survey of Computational Convex Analysis and Its Applications. SIAM Review 52: 505–542. DOI: 10.1137/100788458
Massimo Marinacci and Luigi Montrucchio. 2005. On Concavity and Supermodularity. ICER. Applied Mathematics Working Paper no. 3/2005. On-line at SSRN
Olvi L. Mangasarian. 1968. Characterizations of Real Matrices of Monotone Kind. SIAM Review 10: 439–441. On-line at JSTOR
Olvi L. Mangasarian. 1969. Nonlinear Programming. McGraw–Hill, New York.
Olvi L. Mangasarian. 1979. Simplified Characterizations of Linear Complementarity Problems Solvable as Linear Programs. Mathematics of Operations Research 4: 268–273. On-line at JSTOR
Olvi L. Mangasarian. 1976. Equivalence of the Complementarity Problem to a System of Nonlinear Equations. SIAM Journal of Applied Mathematics 31: 89–92. On-line at JSTOR
Olvi L. Mangasarian. 1981. Iterative Solution of Linear Programs. SIAM Journal on Numerical Analysis 18: 606–614. On-line at JSTOR
Olvi L. Mangasarian. 1994. Nonlinear Programming. SIAM, Philadelphia. Reprint of the 1969 edition published by McGraw–Hill.
P. McMullen. 1970. The maximum numbers of faces of a convex polytope. Mathematika 17-2: 179–184. DOI: 10.1112/S0025579300002850
George J. Minty. 1962. On the Simultaneous Solution of a Certain System of Linear Inequalities. Proceedings of the American Mathematical Society 13: 11–12. On-line at JSTOR
James C. Moore. Some Extensions of the Kuhn–Tucker Results in Concave Programming. In QuirkZarley1968.
Theodore S. Motzkin. 1934. Beiträge Zur Theorie Der Linearen Ungleichungen. It appears from citations I have seen that this was published in Jerusalem in 1936. Motzkin~Motzkin1951Em gives 1934 as the date.
Theodore S. Motzkin. 1951. Two Consequences of the Transposition Theorem on Linear Inequalities. Econometrica 19: 184–185. On-line at JSTOR
Theodore S. Motzkin, Harold Raiffa, G. L. Thompson, and R. M. Thrall. The Double Description Method. In KuhnTucker1953.
Kazuo Murota. 1987. Systems Analysis by Graphs and Matroids: Structural Solvability and Controllability. Springer–Verlag, Berlin, Heidelberg, and New York.
Peter Newman. 1969. Some Properties of Concave Functions. Journal of Economic Theory 1: 291–314. DOI: 10.1016/0022-0531(69)90035-0
Hukukane Nikaidô. 1968. Convex Structures and Economic Theory. Academic Press, New York.
Czeslaw Olech. 1965. A Note Concerning Extremal Points of a Convex Set. Bulletin de l'Académie Polonaise des Sciences; Serie des Sciences Mathématiques, Astronomiques et Physiques 13: 317–321.
Joseph M. Ostroy. 1993. Notes on Linear Programming. Notes for Economics 201C at UCLA.
Josip E. Pečarić, Frank Proschan, and Y. L. Tong. 1992. Convex Functions, Partial Orderings, and Statistical Applications. Academic Press, New York.
T. Pennanen, R. T. Rockafellar, and M. Théra. 2002. Graphical Convergence of Sums of Monotone Mappings. Proceedings of the American Mathematical Society 130: 2261–2269. On-line at JSTOR
Robert R. Phelps. 1993. Convex Functions, Monotone Operators and Differentiability. Springer–Verlag, Berlin.
J. Ponstein. 1967. Seven Kinds of Convexity. SIAM Review 9: 115–119. On-line at JSTOR
J. Ponstein. 1984. Dualizing Optimization Problems in Mathematical Economics. Journal of Mathematical Economics 13: 255–272. DOI: 10.1016/0304-4068(84)90033-8
David Preiss and L. Zajicek. 1984. Frechet Differentiation of Convex Functions in a Banach Space with a Separable Dual. Proceedings of the American Mathematical Society 91: 202–204. On-line at JSTOR
James P. Quirk and Arvid M. Zarley, ed. 1968. Papers in Quantitative Economics, 1. University of Kansas Press, Lawrence, Kansas.
Hans Rademacher and I. J. Schoenberg. 1950. Helly's theorems on convex domains and Tchebycheff's approximation problem. Canadian Journal of Mathematics 2: 245–256. DOI: 10.4153/CJM-1950-022-8
S. Reich. 1972. Fixed Points in Locally Convex Spaces. Mathematische Zeitschrift 125: 17–31.
Juan P. Rincón-Zapatero and Manuel S. Santos. 2009. Differentiability of the value function without interiority assumptions. Journal of Economic Theory 144: 1948–1964. DOI: 10.1016/j.jet.2009.02.006
James W. Roberts. 1977. A Compact Convex Set with No Extreme Points. Studia Mathematica 60: 255–266. On-line at the Polish Virtual Library of Science
A. W. Roberts and D. E. Varberg. 1973. Convex Functions. Academic Press, New York.
A. W. Roberts and D. E. Varberg. 1974. Another Proof that Convex Functions Are Locally Lipschitz. American Mathematical Monthly 81: 1014–1016. On-line at JSTOR
R. T. Rockafellar. 1964. Duality Theorems for Convex Functions. Bulletin of the American Mathematical Society 70: 189-192. On-line at Project Euclid
R. T. Rockafellar. 1964. A Combinatorial Algorithm for Linear Programs in the General Mixed Form. Journal of the Society for Industrial and Applied Mathematics 12: 215–225. On-line at JSTOR
R. T. Rockafellar. 1966. Level Sets and Continuity of Conjugate Convex Functions. Transactions of the American Mathematical Society 123: 46–63. On-line at JSTOR
R. T. Rockafellar. 1968. Integrals Which Are Convex Functionals. Pacific Journal of Mathematics 24: 525–539. On-line at Project Euclid
R. T. Rockafellar. 1970. Convex Analysis. Princeton University Press, Princeton.
R. T. Rockafellar. 1970. On the Maximality of Sums of Nonlinear Monotone Operators. Transactions of the American Mathematical Society 149: 75–88. On-line at JSTOR
R. T. Rockafellar. 1971. Integrals Which Are Convex Functionals. II. Pacific Journal of Mathematics 39: 439–469. On-line at Project Euclid
R. T. Rockafellar. 1971. Existence and Duality Theorems for Convex Problems of Bolza. Transactions of the American Mathematical Society 159: 1–40. On-line at JSTOR
R. T. Rockafellar. 1990. Generalized Second Derivatives of Convex Functions and Saddle Functions. Transactions of the American Mathematical Society 322: 51–77. On-line at JSTOR
R. T. Rockafellar. 1993. Lagrange Multipliers and Optimality. SIAM Review 35: 183–238. On-line at JSTOR
Itsuo Sakuma. 1977. Closedness of convex hulls. Journal of Economic Theory 14: 223–227. DOI: 10.1016/0022-0531(77)90095-3
Rolf Schneider. 1971. On Steiner Points of Convex Bodies. Israel Journal of Mathematics 9: 241–249. DOI: 10.1007/BF02771589
Ivan Singer. 1997. Abstract Convex Analysis. John Wiley and Sons, New York.
Josef Stoer and Christoph Witzgall. 1970. Convexity and Optimization in Finite Dimensions I. Springer–Verlag, Berlin.
Paul R. Thie. 1988. An Introduction to Linear Programming and Game Theory. Wiley, New York.
John W. Tukey. 1942. Some Notes on the Separation of Convex Sets. Portugaliae Mathematicae 3: 95–102. On-line at the Biblioteca Nacional de Portugal
Frederick Valentine. 1964. Convex Sets. McGraw-Hill, New York.
Lieven Vandenberghe and Stephen Boyd. 1996. Semidefinite Programming. SIAM Review 38: 49–95. DOI: 10.1137/1038003
Hermann Weyl. The Elementary Theory of Convex Polyhedra. In KuhnTucker1950.
Menahem E. Yaari. 1977. A Note on Separability and Quasiconcavity. Econometrica 45: 1183–1186. On-line at JSTOR
Zvi Ziegler. 1966. Generalized Convexity Cones. Pacific Journal of Mathematics 17: 561–580. On-line at Project Euclid
Zvi Ziegler. 1968. On the Characterization of Measures of the Cone Dual to a Generalized Convexity Cone. Pacific Journal of Mathematics 24: 603–626. On-line at Project Euclid
Günter M. Ziegler. 1995. Lectures on Polytopes. Springer–Verlag, New York.

Applications

Sydney N. Afriat. 1973. On a System of Inequalities in Demand Analysis: An Extension of the Classical Method. International Economic Review 14: 460–472. On-line at JSTOR
Charalambos D. Aliprantis, KC Border, and W. A. J. Luxemburg, ed. 1991. Positive Operators, Riesz Spaces, and Economics. Springer–Verlag, Berlin.
Robert M. Anderson. 1978. An Elementary Core Equivalence Theorem. Econometrica 46: 1483–1487. On-line at JSTOR
Robert M. Anderson, M. A. Khan, and Salim Rashid. 1982. Approximate Equilibria with Bounds Independent of Preferences. Review of Economic Studies 49: 473–475. On-line at JSTOR
Kenneth J. Arrow and Alain C. Enthoven. 1961. Quasi-Concave Programming. Econometrica 29: 779–800. On-line at JSTOR
Kenneth J. Arrow, Leonid Hurwicz, and Hirofumi Uzawa, ed. 1958. Studies in Linear and Non-Linear Programming. Stanford University Press, Stanford, California.
Itai Ashlagi, Mark Braverman, Avinatan Hassidim, and Dov Monderer. 2011. Monotonicity and Implementability. Manuscript. On-line at MIT
Itai Ashlagi, Mark Braverman, Avinatan Hassidim, and Dov Monderer. 2011. Supplementary Material for Monotonicity and Implementability. Manuscript. On-line at MIT
Claude d'Aspremont, Jacques Cremer, and Louis-André Gerard-Varet. 1993. Correlation, Independence and Bayesian Implementation. SEEDS.
Erik J. Balder. 1998. Lectures on Young Measure Theory and its Applications in Economics. Mathematical Institute, University of Utrecht, Netherlands. On-line
Peter Bardsley. 2011. Duality in Contracting. Department of Economics, University of Melbourne. On-line at Caltech
Aharon Ben-Tal. 1985. The Entropic Penalty Approach to Stochastic Programming. Mathematics of Operations Research 10: 263–279. On-line at JSTOR
David Blackwell. 1953. Equivalent Comparisons of Experiments. Annals of Mathematical Statistics 24: 265–272. On-line at JSTOR
Antoine Billot, Alain Chateauneuf, Itzhak Gilboa, and Jean-Marc Tallon. 2000. Sharing Beliefs: Between Agreeing and Disagreeing. Econometrica 68: 685–694. On-line at JSTOR
Jean-Marc Bonnisseau and Cuong L. Van. 1996. On the Subdifferential of the Value Function in Economic Optimization Problems. Journal of Mathematical Economics 25: 55–73. DOI: 10.1016/0304-4068(95)00717-2
KC Border. 1985. More on Harsanyi's Cardinal Welfare Theorem. Social Choice and Welfare 1: 279–281. DOI: 10.1007/BF00649263
KC Border. Functional Analytic Tools for Expected Utility Theory. In AliprantisBorderLuxemburg1991. On-line at Caltech
KC Border. 1991. Implementation of Reduced Form Auctions: A Geometric Approach. Econometrica 59: 1175–1187. On-line at JSTOR
KC Border. 1992. Revealed Preference, Stochastic Dominance, and the Expected Utility Hypothesis. Journal of Economic Theory 56: 20–42. DOI: 10.1016/0022-0531(92)90067-R
KC Border. 2007. Reduced Form Auctions Revisited. Economic Theory 31: 167–181. DOI: 10.1007/s00199-006-0080-z
T. Börgers. 1994. Weak Dominance and Approximate Common Knowledge. Journal of Economic Theory 64: 265–276.
Donald J. Brown and Caterina Calsimiglia. 2007. The nonparametric approach to applied welfare analysis. Economic Theory 31: 183–188. DOI: 10.1007/s00199-006-0087-5
Donald J. Brown and Rosa L. Matzkin. 1996. Testable Restrictions on the Equilibrium Manifold. Econometrica 64: 1249–1262. On-line at JSTOR
Donald J. Brown and Chris Shannon. 2000. Uniqueness, Stability, and Comparative Statics in Rationalizable Walrasian Markets. Econometrica 68: 1529–1539. On-line at JSTOR
Donald J. Brown and Jan Werner. 1995. Arbitrage and Existence of Equilibrium in Infinite Asset Markets. Review of Economic Studies 62: 101–114. On-line at JSTOR
Michael Bruno. 1969. Fundamental Duality Relations in the Pure Theory of Capital and Growth. Review of Economic Studies 36: 39–53. On-line at JSTOR
Edwin Burmeister and Kiyoshi Kuga. 1970. The Factor-Price Frontier in a Neoclassical Multi-Sector Model. International Economic Review 11: 162–174. On-line at JSTOR
Edwin Burmeister and Kiyoshi Kuga. 1970. The Factor-Price Frontier, Duality and Joint Production. Review of Economic Studies 37: 11–19. On-line at JSTOR
Guillaume Carlier. 2001. A General Existence Result for the Principal-Agent Problem with Adverse Selection. Journal of Mathematical Economics 35: 129–150. DOI: 10.1016/S0304-4068(00)00057-4
Christopher P. Chambers. 2005. Multi-Utilitarianism in Two-Agent Quasilinear Social Choice. International Journal of Game Theory 33: 315–334.
Yeon-Koo Che, Jinwoo Kim, and Konrad Mierendorff. 2011. Generalized Reduced Form Auctions: A Network-Flow Approach. Manuscript.
Yu-Min Chen. 1986. An Extension to the Implementability of Reduced Form Auctions. Econometrica 54: 1249–1251. On-line at JSTOR
Pierre-André Chiappori and Jean-Charles Rochet. 1987. Revealed Preferences and Differentiable Demand. Econometrica 55: 687–691. On-line at JSTOR
John S. Chipman. 1953. Computational Problems in Linear Programming. Review of Economics and Statistics 35: 342–349. On-line at JSTOR
John S. Chipman. 1953. Linear Programming. Review of Economics and Statistics 35: 101–117. On-line at JSTOR
John S. Chipman, Daniel L. McFadden, and Marcel K. Richter, ed. 1990. Preferences, Uncertainty, and Optimality: Essays in Honor of Leonid Hurwicz. Westview Press, Boulder, Colorado.
J.-P. Crouzeix. 1983. Duality between Direct and Indirect Utility Functions: Differentiability Properties. Journal of Mathematical Economics 12: 149–165. DOI: 10.1016/0304-4068(83)90010-1
Jakša Cvitanić and Ioannis Karatzas. 1992. Convex Duality in Constrained Portfolio Optimization. Annals of Applied Probability 2: 767–818. On-line at JSTOR
W. E. Diewert. 1971. An Application of the Shephard Duality Theorem: A Generalized Leontief Production Function. Journal of Political Economy 79: 481–507. On-line at JSTOR
W. E. Diewert. 1973. Afriat and Revealed Preference Theory. Review of Economic Studies 40: 419–425. On-line at JSTOR
W. E. Diewert and Alan D. Woodland. 1977. Frank Knight's Theorem in Linear Programming Revisited. Econometrica 45: 375–398. On-line at JSTOR
Lloyd L. Dines. 1917. Concerning Preferential Voting. American Mathematical Monthly 24: 321–325. On-line at JSTOR
Robert Dorfman, Paul A. Samuelson, and Robert M. Solow. 1987. Linear Programming and Economic Analysis. Dover, New York. Reprint of the 1958 edition published by McGraw-Hill in New York.
Philip H. Dybvig and Stephen A. Ross. 1982. Portfolio Efficient Sets. Econometrica 50: 1525–1546. On-line at JSTOR
Edmund Eisenberg and David Gale. 1959. Consensus of Subjective Probabilities: The Pari-Mutuel Method. Annals of Mathematical Statistics 30: 165–168. On-line at JSTOR
Peter C. Fishburn. 1973. The Theory of Social Choice. Princeton University Press, Princeton.
Peter C. Fishburn. 1973. Summation Social Choice Functions. Econometrica 41: 1183–1196. On-line at JSTOR
Peter C. Fishburn. 1974. Convex Stochastic Dominance with Continuous Distribution Functions. Journal of Economic Theory 7: 143–158. DOI: 10.1016/0022-0531(74)90103-3
Peter C. Fishburn. 1974. Majority Voting on Risky Investments. Journal of Economic Theory 8: 85–99. DOI: 10.1016/0022-0531(74)90007-6
Peter C. Fishburn. 1975. Separation Theorems and Expected Utility. Journal of Economic Theory 11: 16–34. DOI: 10.1016/0022-0531(75)90036-8
David A. Freedman and Roger A. Purves. 1969. Bayes' Method for Bookies. Annals of Mathematical Statistics 40: 1177–1186. On-line at JSTOR
Satoru Fujishige and Akihisa Tamura. 2005. A Two-Sided Discrete-Concave Market with Possibly Bounded Side Payments: An Approach by Discrete Convex Analysis. Kyoto University. RIMS Preprint no. 1470.
David Gale. 1960. Theory of Linear Economic Models. McGraw-Hill, New York.
David Gale. 1967. A Geometric Duality Theorem with Economic Applications. Review of Economic Studies 34: 19–24. On-line at JSTOR
David Gale. 1973. On the Theory of Interest. American Mathematical Monthly 80: 853–868. On-line at JSTOR
Itzhak Gilboa and David Schmeidler. 1989. Maximin Expected Utility with Non-Unique Prior. Journal of Mathematical Economics 18: 141–153. DOI: 10.1016/0304-4068(89)90018-9
Arlo D. Hendrickson and Robert J. Buehler. 1971. Proper Scores for Probability Forecasters. Annals of Mathematical Statistics 42: 1916–1921. On-line at JSTOR
David C. Heath and William D. Sudderth. 1972. On a Theorem of De Finetti, Oddsmaking, and Game Theory. Annals of Mathematical Statistics 43: 2072–2077. On-line at JSTOR
Leonid Hurwicz. Programming in Linear Spaces. In ArrowHurwiczUzawa1958.
Leonid Hurwicz, David Schmeidler, and Hugo F. Sonnenschein, ed. 1985. Social Goals and Social Organization: Essays in Honor of Elisha Pazner. Cambridge University Press, Cambridge.
Dale W. Jorgenson and Lawrence J. Lau. 1974. The Duality of Technology and Economic Behaviour. Review of Economic Studies 41: 181–200. On-line at JSTOR
Robert P. Kertz and Uwe Rösler. 1992. Stochastic and Convex Orders and Lattices of Probability Measures, with a Martingale Interpretation. Israel Journal of Mathematics 77: 129–164. DOI: 10.1007/BF02808015
Taesung Kim. 1991. The Subjective Expected Utility Hypothesis and Revealed Preference. Economic Theory 1: 251–263. DOI: 10.1007/BF01210563
Taesung Kim and John O. Ledyard. 1994. First Best Bayesian Privatization Mechanisms. Manuscript.
Tjalling C. Koopmans, ed. 1951. Activity Analysis of Production and Allocation: Proceedings of a Conference. John Wiley and Sons, New York. On-line at the Cowles Commission
Tjalling C. Koopmans. 1953. Activity Analysis and Its Applications. American Economic Review 43: 406–414. On-line at JSTOR
Tjalling C. Koopmans. 1961. Convexity Assumptions, Allocative Efficiency, and Competitive Equilibrium. Journal of Political Economy 69: 478–479. On-line at JSTOR
Tjalling C. Koopmans. 1977. Concepts of Optimality and Their Uses. American Economic Review 67: 261–274. On-line at JSTOR
Victor L. Klee, Jr. and George G. Minty. How Good is the Simplex Algorithm? In Shisha1972.
Charles H. Kraft, John W. Pratt, and A. Seidenberg. 1959. Intuitive Probability on Finite Sets. Annals of Mathematical Statistics 30: 408–419. On-line at JSTOR
David H. Krantz, R. D. Luce, Patrick Suppes, and Amos Tversky. 1971. Foundations of Measurement. Academic Press, New York.
David M. Kreps. 1979. A Representation Theorem for ``Preference for Flexibility''. Econometrica 47: 565–578. On-line at JSTOR
David M. Kreps. 1981. Arbitrage and Equilibrium in Economies with Infinitely Many Commodities. Journal of Mathematical Economics 8: 15–35. DOI: 10.1016/0304-4068(81)90010-0
Harold W. Kuhn and Albert W. Tucker, ed. 1950. Contributions to the Theory of Games, I. Princeton University Press, Princeton.
John O. Ledyard. 1986. The Scope of the Hypothesis of Bayesian Equilibrium. Journal of Economic Theory 39: 59–82. DOI: 10.1016/0022-0531(86)90020-7
R. D. Luce. 1967. Sufficient Conditions for the Existence of a Finitely Additive Probability Measure. Annals of Mathematical Statistics 38: 780–786. On-line at JSTOR
Fabio Maccheroni, Massimo Marinacci, and Aldo Rustichini. 2006. Ambiguity Aversion, Robustness, and the Variational Representation of Preferences. Econometrica 74: 1447–1498. On-line at JSTOR
Mark J. Machina and David Schmeidler. 1992. A More Robust Definition of Subjective Probability. Econometrica 60: 745–780. On-line at JSTOR
Edmond Malinvaud. 1953. Capital Accumulation and Efficient Allocation of Resources. Econometrica 21: 233–268. On-line at JSTOR
Edmond Malinvaud. 1962. Efficient Capital Accumulation: A Corrigendum. Econometrica 30: 570–573. On-line at JSTOR
Alan S. Manne, Hung-Po Chao, and Robert Wilson. 1980. Computation of Competitive Equilibria by a Sequence of Linear Programs. Econometrica 48: 1595–1615. On-line at JSTOR
Juan-Enrique Martínez-Legaz and Manuel S. Santos. 1996. On expenditure functions. Journal of Mathematical Economics 25: 143–163. DOI: 10.1016/0304-4068(95)00725-3
Andreu Mas-Colell. 1976. A Remark on a Smoothness Property of Convex, Complete Preorders. Journal of Mathematical Economics 3: 103–105. DOI: 10.1016/0304-4068(76)90008-2
Rosa L. Matzkin and Marcel K. Richter. 1991. Testing Strictly Concave Rationality. Journal of Economic Theory 53: 287–303. DOI: 10.1016/0022-0531(91)90157-Y
Daniel L. McFadden and Marcel K. Richter. Stochastic Rationality and Revealed Preference. In ChipmanMcFaddenRichter1990.
Konrad Mierendorff. 2011. Asymmetric Reduced Form Auctions. Economics Letters 110: 41–44. DOI: 10.1016/j.econlet.2010.09.019
Paul R. Milgrom. 1981. Good News and Bad News: Representation Theorems and Applications. Bell Journal of Economics 12: 380–391. On-line at JSTOR
Michael Mussa and Sherwin Rosen. 1978. Monopoly and Product Quality. Journal of Economic Theory 18: 301–317. DOI: 10.1016/0022-0531(78)90085-6
Roger B. Myerson. 1979. Incentive Compatibility and the Bargaining Problem. Econometrica 47: 61–74. On-line at JSTOR
Roger B. Myerson. 1981. Optimal Auction Design. Mathematics of Operations Research 6: 58–73. On-line at JSTOR
Roger B. Myerson. Bayesian Equilibrium and Incentive-Compatibility: An Introduction. In HurwiczSchmeidlerSonnenschein1985.
Takashi Negishi. 1960. Welfare Economics and Existence of an Equilibrium for a Competitive Economy. Metroeconomica 12: 92–97. DOI: 10.1111/j.1467-999X.1960.tb00275.x
Hukukane Nikaidô. 1968. Convex Structures and Economic Theory. Academic Press, New York.
Yoshihiko Otani. 1973. Neo-Classical Technology Sets and Properties of Production Possibility Sets. Econometrica 41: 667–682. On-line at JSTOR
Kiyoshi Otani. 1983. A characterization of quasi-concave functions. Journal of Economic Theory 31: 194–196. DOI: 10.1016/0022-0531(83)90030-3
Bezalel Peleg. 1970. Efficiency Prices for Optimal Consumption Plans (I). Journal of Mathematical Analysis and Applications 29: 83–90. DOI: 10.1016/0022-247X(70)90102-2
Bezalel Peleg. 1971. Efficiency Prices for Optimal Consumption Plans (II). Israel Journal of Mathematics 9: 222–234. DOI: 10.1007/BF02771587
Bezalel Peleg. 1970. Efficiency Prices for Optimal Consumption Plans (III). Journal of Mathematical Analysis and Applications 32: 630–638. DOI: 10.1016/0022-247X(70)90286-6
Bezalel Peleg and Menachem E. Yaari. 1970. Efficiency Prices in Infinite-Dimensional Space. Journal of Economic Theory 2: 41–85.
James P. Quirk and Rubin Saposnik. 1966. Homogeneous Production Functions and Convexity of the Production Possibility Set. Metroeconomica 18: 192–197. DOI: 10.1111/j.1467-999X.1966.tb00855.x
Marcel K. Richter and Kam-Chau Wong. 2004. Concave Utility on Finite Sets. Journal of Economic Theory 115: 341–357. DOI: 10.1016/S0022-0531(03)00167-4
Jean-Charles Rochet. 1987. A Necessary and Sufficient Condition for Rationalizability in a Quasi-Linear Context. Journal of Mathematical Economics 16: 191–200. DOI: 10.1016/0304-4068(87)90007-3
Dov Samet. 1998. Common Priors and the Separation of Convex Sets. Games and Economic Behavior 24: 172–174. DOI: 10.1006/game.1997.0615
Dov Samet. 1998. Iterated Expectations and Common Priors. Games and Economic Behavior 24: 131–141. DOI: 10.1006/game.1997.0616
Paul A. Samuelson. 1962. Parable and Realism in Capital Theory: The Surrogate Production Function. Review of Economic Studies 29: 193–206. On-line at JSTOR
Dana Scott. 1964. Measurement Structures and Linear Inequalities. Journal of Mathematical Psychology 1: 233–247. DOI: 10.1016/0022-2496(64)90002-1
Oved Shisha, ed. 1972. Inequalities III. Academic Press, New York.
B. P. Stigum. 1968. On a Property of Concave Functions. Review of Economic Studies 35: 413–416. On-line at JSTOR
Akira Takayama and Mohamed El-Hodiri. 1968. Programming, Pareto Optimum and the Existence of Competitive Equilibria. Metroeconomica 20: 1–10. DOI: 10.1111/j.1467-999X.1968.tb00117.x
T. Takayama and Alan D. Woodland. 1970. Equivalence of Price and Quantity Formulations of Spatial Equilibrium: Purified Duality in Quadratic and Concave Programming. Econometrica 38: 889–906. On-line at JSTOR
Arja H. Turunen-Red and Alan D. Woodland. 1994. On Economic Applications of the Kuhn–Fourier Theorem. Manuscript.
Arja H. Turunen-Red and Alan D. Woodland. On Economic Applications of the Kuhn–Fourier Theorem. In Wooders1999.
Hirofumi Uzawa. The Kuhn–Tucker Conditions in Concave Programming. In ArrowHurwiczUzawa1958.
Hal R. Varian. 1987. The Arbitrage Principle in Financial Economics. Journal of Economic Perspectives 1: 55–72. On-line at JSTOR
Xavier Vives. 1990. Nash Equilibrium with Strategic Complementarities. Journal of Mathematical Economics 19: 305–321. DOI: 10.1016/0304-4068(90)90005-T
Hermann Weyl. Elementary Proof of a Minimax Theorem Due to von Neumann. In KuhnTucker1950.
Andrew Whinston. 1965. Conjugate functions and dual programs. Naval Research Logistics Quarterly 12: 315–322. DOI: 10.1002/nav.3800120307
G. A. Whitmore. 1975. The Theory of Skewness Preference. Journal of Business Administration 6: 13–20.
Myrna H. Wooders, ed. 1999. Topics in Mathematical Economics and Game Theory: Essays in Honor of Robert J. Aumann. American Mathematical Society, Providence, RI.

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