Semi-Infinite Programming pp 3-41 | Cite as

# On the 1962–1972 Decade of Semi-Infinite Programming: A Subjective View

## Abstract

Several major themes developed during this, the apparent first of almost four decades of semi-infinite programming, are reviewed in this paper.

One theme was the development of a dual program to the problem of minimizing an arbitrary convex function over an arbitrary convex set in the *n*-space that featured the maximization of a linear functional in non-negative variables of a generalized finite sequence space subject to a finite system of linear inequalities. A characteristic of the dual program was that it did not involve any primal variables occurring within an internal optimization.

A second major theme was the introduction of an “infinity” into systems of semi-infinite linear inequalities, a manifestation of the “probing” between analysis and algebra.

In finite linear programming there are four mutually exclusive and collectively exhaustive duality states that can occur, and this led to the third theme of developing a classification theory for linear semi-infinite programming that included finite linear programming as a special case.

The fourth theme was one of algorithmic development. Finally, throughout the decade there was an emphasis on applications, principally to Economics, Game Theory, Asymptotic Lagrange Regularity, Air Pollution Abatement, and Geometric Programming.

## Keywords

Linear Inequality Geometric Programming Duality State Dual Program Subjective View## Preview

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## References

- [1]K.J. Arrow and L. Hurwicz. Decentralization and computation in resource allocation. In P. W. Pfouts, editor,
*Essays in Economics and Econometrics*, pages 34–104, University of North Carolina, Chapel Hill, N. C, 1960.Google Scholar - [2]W. J. Baumol and T. Fabian. Decomposition, pricing for decentralization and external economies,
*Management Science*, 11:241–261, 1964.CrossRefGoogle Scholar - [3]E. J. Beltrami. A constructive proof of the Kuhn-Tucker multiplier rule,
*Journal of Mathematical Analysis and Applications*, 26:297–306, 1967.MathSciNetCrossRefGoogle Scholar - [4]A. Ben-Israel, A. Charnes, and K. O. Kortanek. Duality and asymptotic solvability over cones,
*Bulletin of the American Mathematical Society*,75:318–324, 1969.MathSciNetCrossRefzbMATHGoogle Scholar - [4a]A. Ben-Israel, A. Charnes, and K. O. Kortanek. Erratum,
*Bulletin of the American Mathematical So ciety*76 (1970) 428.Google Scholar - [5]A. Ben-Israel, A. Charnes, and K. O. Kortanek. Asymptotic duality in semi-infinite programming and the convex core topology,
*Rendiconti Di Matematica (Rome)*, 4:751–761, 1971. Presented at the Conference on Mapping Techniques and Problems University of Houston, November, 1970 in honor of Professor David Bourgin.MathSciNetGoogle Scholar - [6]J. H. Bigelow, J. C. DeHaven, and N. Z. Shapiro. Chemical equilibrium problems with unbounded solution sets,
*SIAM Journal on Applied Math ematics*, 18:768–775, 1970.MathSciNetCrossRefGoogle Scholar - [7]O. N. Bondareva. Nekotorye primeneniia metodor linejnogo program mirovaniia k teorii kooperativnykh igr.,
*Problemy Kibernetiki*, 10:119–139, 1963. Translation of Title: Some applications of linear programming methods to the theory of cooperative games.MathSciNetGoogle Scholar - [8]A. Charnes, R. W. Clower, and K. O. Kortanek. Effective control through coherent decentralization with pre-emptive goals,
*Econometrica*, 35:294–320, 1967.CrossRefzbMATHGoogle Scholar - [9]A. Charnes and W. W. Cooper. The strong Minkowski-Farkas-Weyl theorem for vector spaces over ordered fields, [
*S. Proceedings of the National Academic of Sciences*, 44:914–916, 1969.MathSciNetCrossRefGoogle Scholar - [10]A. Charnes and W. W. Cooper.
*Management Models and Industrial Applications of Linear Programming, Volumes I and II*, Wiley, 1961.Google Scholar - [11]A. Charnes, W. W. Cooper, and K. O. Kortanek. Duality, Haar programs and finite sequence spaces,
*U*. S.*Proceedings of the National Academy of Sciences*, 48:782–786, 1962.MathSciNetCrossRefGoogle Scholar - [12]A. Charnes, W. W. Cooper, and K. O. Kortanek. A duality theory for convex programs with convex constraints,
*Bulletin of the American Mathematical Society*, 68:605–608, 1962.MathSciNetCrossRefzbMATHGoogle Scholar - [13]A. Charnes, W. W. Cooper, and K. O. Kortanek. Duality in semi-infinite programs and some works of Haar and Caratheodory,
*Management Science*, 9:208–228, 1963.MathSciNetGoogle Scholar - [14]A. Charnes, W. W. Cooper, and K. O. Kortanek. On representation of semi-infinite programs which have no duality gaps,
*Management Science*, 12:113–121, 1965.MathSciNetCrossRefzbMATHGoogle Scholar - [15]A. Charnes, W. W. Cooper, and K. O. Kortanek. On some nonstandard semi-infinite programming problems. Technical report No. 45, Cornell University, Department of Operations Research, Ithaca, N. Y., March 1968.Google Scholar
- [16]A. Charnes, W. W. Cooper, and K. O. Kortanek. On the theory of semi infinite programming and some generalizations of Kuhn-Tucker saddle point theorems for arbitrary convex functions,
*Naval Research Logistics Quarterly*, 16:41–51, 1969.MathSciNetzbMATHGoogle Scholar - [17]A. Charnes, W. W. Cooper, and K. O. Kortanek. Semi-infinite program ming, differentiability, and geometric programming Part II,
*Aplikace Matematicky (Prague)*, 14:15–22, 1969.MathSciNetzbMATHGoogle Scholar - [18]A. Charnes, W. W. Cooper, and K. O. Kortanek. Semi-infinite program ming, differentiability, and geometric programming,
*Journal of Mathe matical Sciences*, 6:19–40), 1971. R. S. Varma Memorial Volume.MathSciNetGoogle Scholar - [19]A. Charnes, M. J. Eisner, and K. O. Kortanek. On weakly balanced games and duality theory,
*Cahiers du Centre d’Etude de Recherche Opéra tionnelle (Belgium)*, 12:7–21, 1970.MathSciNetzbMATHGoogle Scholar - [20]A. Charnes and K. O. Kortanek. An opposite sign algorithm for purifica tion to an extreme point solution. O. N. R. Research Memorandum No. 84, Northwestern University, The Technological Institute, Evanston, Illinois, June 1963.Google Scholar
- [21]A. Charnes and K. O. Kortanek. On balanced sets, cores, and linear programming.
*Cahiers du Centre d’Etude de Recherche Opérationnelle (Belgium)*, 9:32–43, 1967.MathSciNetzbMATHGoogle Scholar - [22]A. Chaînes and K. O. Kortanek. On the status of separability and non separability in decentralization theory,
*Management Science: Applica tions*, 15:B12-B14, 1968.Google Scholar - [23]A. Charnes and K. O. Kortanek. On classes of convex and preemptive nuclei for n-person games. In H. W. Kuhn, editor,
*Proceedings of the Princeton Symposium on Mathematical Programming*, pages 377–390. Mathematical Programming Society, Princeton University Press, 1970.Google Scholar - [24]C. K. Chui and J. D. Ward. Book review of “Optimization and Approxi mation” by W. Krabs, Wiley, 1979.
*Bulletin of the American Mathemat ical Society*, 3:1056–1069, 1980.MathSciNetGoogle Scholar - [25]G. B. Dantzig.
*Linear Programming and Extensions*, Princeton University Press, 1963.zbMATHGoogle Scholar - [26]O. A. Davis and K. O. Kortanek. Centralization and decentralization: the political economy of public school systems,
*American Economic Review*, 61:456–462, 1971.Google Scholar - [27]G. Debreu.
*Theory of Value*, Wiley, 1959. Cowles Foundation for Research in Economics at Yale University, Monograph 17. Copyright renewed 1987 for the Yale University Press.zbMATHGoogle Scholar - [28]U. Dieter. Optimierungsaufgaben in topologischen Vektorräumen I: Du alitätsheorie, Z.
*Wahrscheinlichkeitstheorie verw*, 5:89–117, 1966.MathSciNetCrossRefzbMATHGoogle Scholar - [29]W. S. Dorn. Duality in quadratic programming,
*Quarterly Journal of Applied Mathematics*, 20:155–162, 1960.MathSciNetGoogle Scholar - [30]R. J. Duffin. Infinite programs, In H. W. Kuhn and A. W. Tucker, edi tors,
*Linear Inequalities and Related Systems*, pages 157–170. Princeton University Press, 1956.Google Scholar - [31]R. J. Duffin. Dual programs with minimum cost, 7.
*Society on Industrial & Applied Mathematics*, 10:119–123, 1962.MathSciNetCrossRefzbMATHGoogle Scholar - [32]R. J. Duffin. An orthogonality theorem of Dines related to moment prob lems and linear programming,
*Journal of Combinatorial Theory*, 2:1–26, 1967.MathSciNetCrossRefzbMATHGoogle Scholar - [33]R. J. Duffin. Duality inequalities of mathematics and science, In
*Nonlinear Programming*, pages 401–23. Academic Press, 1970.Google Scholar - [34]R. J. Duffin. Linearizing geometric programs,
*SIAMReview*, 12:211–227, 1970.MathSciNetCrossRefzbMATHGoogle Scholar - [35]R. J. Duffin and L. A. Karlovitz. An infinite linear program with a duality gap,
*Management Science*, 12:122–134, 1965.MathSciNetCrossRefzbMATHGoogle Scholar - [36]R. J. Duffin, E. L. Peterson, and C. L. Zener.
*Geometric Programming Theory and Applications*, Wiley, 1967.Google Scholar - [37]E. Eisenberg. Duality in homogeneous programming,
*Proceedings of the American Mathematical Society*, 12:783–787, 1961.MathSciNetCrossRefzbMATHGoogle Scholar - [38]E. Eisenberg. Supports of a convex function,
*Bulletin of the American Mathematical Society*, 68:192, 1961.MathSciNetCrossRefGoogle Scholar - [39]J. P. Evans. Duality in Markov decision problems with countable action and state spaces,
*Management Science*, 15:626–638, 1969.CrossRefzbMATHGoogle Scholar - [40]J. P. Evans and K. O. Kortanek. Pseudo-concave programming and La grange regularity,
*Operations Research*, 15:882–891, 1967.MathSciNetCrossRefzbMATHGoogle Scholar - [41]J. P. Evans and K. O. Kortanek. Asymptotic Lagrange regularity for pseudo-concave programming with weak constraint qualification,
*Oper ations Research*, 16:849–857, 1968.MathSciNetCrossRefzbMATHGoogle Scholar - [42]K. Fan. Asymptotic cones and duality of linear relations,
*Journal of Approximation Theory*, 2:152–159, 1969.MathSciNetCrossRefzbMATHGoogle Scholar - [43]A. V. Fiacco and G. P. McCormick. Asymptotic conditions for constrained minimization. Technical Report RAC-TP-340, Research Analysis Corpo ration, McLean, Virginia, 1968. The firm no longer exists.Google Scholar
- [44]D. Gale. A geometric duality theorem with economic applications,
*Review of Economic Studies*, 34:19–24, 1967.CrossRefGoogle Scholar - [45]W. Gochet.
*Computational Treatment of Some Linear Programming Op timization Problems with Applications to Geometric Programming and Probabilistic Programming*. PhD thesis, Carnegie Mellon University, Graduate School of Industrial Administration, Pittsburgh, Pennsylvania, June 1972.Google Scholar - [46]W. Gochet, K. O. Kortanek, and Y. Smeers. On a classification scheme for geometric programming and complementarity. Technical report, Carnegie-Mellon University, Graduate School of Industrial Administra tion, Pittsburgh, Pennsylvania, October 1971.Google Scholar
- [47]W. Gochet and Y. Smeers. On the use of linear programs to solve prototype geometric programs. CORE Discussion Paper No. 7229, University of Louvain, Brussels, Belgium, November 1972.Google Scholar
- [48]E. G. Gol’stein.
*Theory of Convex Programming*, Translations of Mathe matical Monographs, Vol. 36 of American Mathematical Society, Provi dence, 1972.Google Scholar - [49]W. L. Gorr, S. -Å.Gustafson, and K. O. Kortanek. Optimal control strate gies for air quality standards and regulatory policy,
*Environment and Planning*, 4:183–192, 1972.CrossRefGoogle Scholar - [50]H. J. Greenberg. Mathematical programming models for environmental quality control,
*Operations Research*, 43:578–622, 1995.CrossRefzbMATHGoogle Scholar - [51]M. Guignard. Generalized Kuhn-Tucker conditions for mathematical pro gramming problems in a Banach space,
*SIAM Journal on Control*, 7:232 241, 1969.MathSciNetGoogle Scholar - [52]S. -A. Gustafson. On the computational solution of a class of generalized moment problems,
*SIAM Journal on Numerical Analysis*, 7:343–357, 1970.MathSciNetCrossRefzbMATHGoogle Scholar - [53]S. -A. Gustafson. Nonlinear systems in semi-infinite programming, Se ries in Numerical Optimization & Pollution Abatement Technical Report No. 2, Carnegie Mellon University, School of Urban and Public Affairs, Pittsburgh, Pennsylvania, July 1972.Google Scholar
- [54]S. -À. Gustafson and K. O. Kortanek. Numerical treatment of a class of semi-infinite programming problems, Institute of Physical Planning Technical Report No. 21, Carnegie Mellon University, School of Urban and Public Affairs, Pittsburgh, Pennsylvania, August 1971.Google Scholar
- [55]S. -Â. Gustafson and K. O. Kortanek. Analytical properties of some multiple-source urban diffusion models,
*Environment and Planning*, 4:31–41, 1972.CrossRefGoogle Scholar - [56]S. -X. Gustafson and K. O. Kortanek. Numerical treatment of a class of convex problems, Series in Numerical Optimization & Pollution Abate ment Technical Report No. 4, Carnegie Mellon University, School of Ur ban and Public Affairs, Pittsburgh, Pennsylvania, July 1972.Google Scholar
- [57]S. -À. Gustafson, K. O. Kortanek, and W. O. Rom. Non-chebysevian moment problems,
*SIAM Journal on Numerical Analysis*, 7:335–342, 1970.MathSciNetCrossRefzbMATHGoogle Scholar - [58]A. Haar. Xüber lineare Ungleichungen,
*Acta Universitatis Szegedienis*, 2:1–14, 1924.zbMATHGoogle Scholar - [59]G. H. Hardy.
*Orders of Infinity*, Cambridge Tracts in Mathematics and Mathematical Physics. Cambridge University Press, Cambridge, U. K., 1954.Google Scholar - [60]G. H. Hardy.
*A Course of Pure Mathematics*, Cambridge University Press, U. K., tenth edition, 1958.Google Scholar - [61]R. G. Jeroslow and K. O. Kortanek. Algebraic Hubert field characteriza tions of asymptotic duality states and optimal paths to infinity, Technical report, Carnegie-Mellon University, Graduate School of Industrial Admin istration, Pittsburgh, Pennsylvania, August 1970. Management Sciences Research Report No. 215.Google Scholar
- [62]R. G. Jeroslow and K. O. Kortanek. On semi-infinite systems of linear inequalities,
*Israel Journal of Mathematics*, 10:252–258, 1971.MathSciNetCrossRefzbMATHGoogle Scholar - [63]C. Kallina and A. C. Williams. Duality and solvability theorems over cones, Technical report, Mobil Research and Development Corporation, Princeton, N. J., August 1969.Google Scholar
- [64]C. Kallina and A. C. Williams. Linear programming in reflexive spaces,
*SIAM Review*, 13:350–376, 1971.MathSciNetCrossRefzbMATHGoogle Scholar - [65]S. Karlin and W. S. Studden.
*Tchebychev Systems: with Applications in Analysis and Statistics*, Wiley, 1966.Google Scholar - [66]K. O. Kortanek.
*Duality, Semi-Infinite Programming and Some Aspects of Control in Business and Economic Systems*, PhD thesis, Northwestern University, Field of Engineering Science, Evanston, Illnois, June 1964.Google Scholar - [67]K. O. Kortanek. Compound asymptotic duality classification schemes, Technical report, Carnegie-Mellon University, Graduate School of Indus trial Administration, Pittsburgh, Pennsylvania, November 1969. Manage ment Sciences Research Report No. 185.Google Scholar
- [68]K. O. Kortanek. Effective control through coherent decentralization in separably and non-separably structured organizations, In R. Chisholm, M. Radnor, and M. F. Tuite, editors,
*Interorganizational Decision Making*, pages 70–82. Aldine (Chicago), 1972.Google Scholar - [69]K. O. Kortanek. On a compound duality classification scheme with homo geneous dérivants,
*Rendiconti Di Maternatica (Rome)*, 5:349–356, 1972.MathSciNetzbMATHGoogle Scholar - [70]K. O. Kortanek and J. P. Evans. On the’M-operator’ and redundant inequalities of the core of a game. Technical report No. 43, Cornell Uni versity, Department of Operations Research, Ithaca, New York, February 1968.Google Scholar
- [71]K. O. Kortanek and W. L. Gorr. Numerical aspects of pollution abatement problems: optimal control strategies for air quality standards. In M. Henke, A. Jaeger, R. Wartmann, and J. H. Zimmerman, editors,
*Proceedings in Operations Research*, pages 34–58. Physic-Verlag (Wurzburg-Wien), 1972.Google Scholar - [72]K. O. Kortanek and W. O. Rom. Classification schemes for the strong duality of linear programming over cones,
*Operations Research*, 19:1571–1585, 1971.MathSciNetCrossRefzbMATHGoogle Scholar - [73]K. O. Kortanek and A. L. Soyster. On classification schemes of some solution sets of chemical equilibrium problems. Institute of Physical Planning Technical Report No. 6, Carnegie Mellon University, School of Urban and Public Affairs, Pittsburgh, Pennsylvania, July 1970.Google Scholar
- [74]K. O. Kortanek and A. L. Soyster. On refinements of some duality theorems in linear programming over cones,
*Operations Research*, 20:137–142, 1972.MathSciNetCrossRefzbMATHGoogle Scholar - [75]W. Krabs.
*Optimierung und Approximation*. B. G.Teubner, Stuttgart, Germany, 1979.Google Scholar - [76]K. S. Kretschmer. Programmes in paired spaces,
*Canadian Journal of Mathematics*, 13:221–238, 1961.MathSciNetCrossRefzbMATHGoogle Scholar - [77]R. T. Rockafellar. Duality theorems for convex functions,
*Bulletin of the American Mathematical Society*, 70:189–192, 1964.MathSciNetCrossRefzbMATHGoogle Scholar - [78]R. T. Rockafellar.
*Convex Analysis*. Princeton University Press, 1970.zbMATHGoogle Scholar - [79]W. O. Rom.
*Classification Theory in Mathematical Programming and Applications*. PhD thesis, Cornell University, Industrial Engineering and Operations Research, Ithaca, N. Y., June 1970.Google Scholar - [80]M. Schechter. Linear programs in topological vector spaces. Technical report, Lehigh University, Bethlehem, Pennsylvania 18015, 1970. College of Arts and Science Report.Google Scholar
- [81]D. Schmeidler. On balanced games with infinitely many players. Research memorandum no. 28, Hebrew University, Department of Mathematics, Jerusalem, Israel, 1967.Google Scholar
- [82]L. S. Shapley. On balanced sets and cores,
*Naval Research Logistics Quarterly*, 14:32–43, 1967.CrossRefGoogle Scholar - [83]Y. Smeers.
*Geometrie Programming and Management Science*. PhD thesis, Carnegie Mellon University, Graduate School of Industrial Ad ministration, Pittsburgh, Pennsylvania, June 1972.Google Scholar - [84]R. E. Train and J. Carroll. Environmental Management and Mathematics,
*SIAM News*, 7:2–3, 1974. In 1974 Russell Train was the Administrator of the Federal Environmental Protection Agency (EPA) and James Carroll was a member of Office and Planning Management at the EPA.Google Scholar - [85]S. N. Tschernikow. O teoreme chaara dlja beskonetschnych sistem line jnych neravenctv,
*Uspechi matern. nauk*, 113:199–200, 1963. Translation of Title: On Haar’s theorem about infinite systems of linear inequalities. The journal Uspechi matem. nauk translates into Successes in Math. Sci ences. As far we know, there exists no English translation of this journal.Google Scholar - [86]S. N. Tschernikow. Poliedraljno samknutye sistemy linejnych neravenstv,
*DokladyAkad. NaukSSSR*, 161:55–58, 1965.Google Scholar - [86]S. N. Tschernikow. Translation: Polyhedrally closed systems of linear inequalities. Soviet Math. Doklady, 6:381–384.Google Scholar
- [87]S. N. Tschernikow.
*Lineare Ungleichungen*, chapter 7. Deutscher Verlag der Wissenschaften Berlin, 1971. Translation from the Russian Linejnye neravenstva (Linear inequalities) published 1968 by Nauka in Moscow.zbMATHGoogle Scholar - [88]A. Whinston.
*Price Coordination in Decentralized Systems*. PhD thesis, Carnegie Mellon University, Graduate School of Industrial Administra tion, Pittsburgh, Pennsylvania, June 1962. Office of Naval Research Memo No. 99.Google Scholar - [89]A. Whinston. Price guides in decentralized organizations. In W. W. Cooper, H. J. Levitt, and M. W. Shelly, editors,
*New Perspectives in Or ganizational Research*. Wiley, 1964.Google Scholar - [90]A. Whinston. Some Applications of the Conjugate Function Theory to Duality, In J. Abadie, editor,
*Nonlinear Programming*, chapter 5. Wiley, 1967.Google Scholar - [91]Y. J. Zhu. Generalizations of some fundamental theorems on linear in equalities,
*Acta Mathematicae Sinica*, 16:25–40, 1966.zbMATHGoogle Scholar - [92]S. Zlobec. Asymptotic Kuhn-Tucker conditions for mathematical pro gramming problems in a Banach space,
*SIAM Journal on Control*, 8:505–512, 1970.MathSciNetCrossRefzbMATHGoogle Scholar - [93]S. Zlobec. Extensions of asymptotic Kuhn-Tucker conditions in mathe matical programming,
*SIAM Journal on Applied Mathematics*, 21:448–460, 1971.MathSciNetCrossRefzbMATHGoogle Scholar