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A survey of vector optimization in infinite-dimensional spaces, part 2

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Abstract

The present survey deals with the state of vector optimization as a mathematical discipline. In this context, the optima are generally defined as maximal pointsy 0 with respect to a partial order on the criteria space. The survey is restricted to a discussion of that literature which deals with pointsy 0 which satisfy a maximality condition with respect toy 0-comparable criteria values; papers which are based on a maximality condition satisfied for all admissible criteria values are included only in a supplementary bibliography. For the former, all aspects of the optimization process are surveyed, ranging from questions of existence to the treatment of duality. Particular attention is paid to questions of proper maximality. The discussion is based on a broad range of definitions and selected theorems from the literature.

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References

  1. Hurwicz, L.,Programming in Linear Spaces, Studies in Linear and Nonlinear Programming, Edited by K. J. Arrow, L. Hurwicz, and H. Uzawa, Stanford University Press, Stanford, California, 1958; Second Printing, Oxford University Press, London, England, pp. 38–102, 1964.

    Google Scholar 

  2. Stadler, W.,A Survey of Multicriteria Optimization or the Vector Maximal Problem, Part 1: 1776–1960, Journal of Optimization Theory and Applications, Vol. 29, pp. 1–52, 1979.

    Google Scholar 

  3. Stadler, W.,A Comprehensive Bibliography on Multicriteria Decision Making and Related Areas, Department of Mechanical Engineering Report, University of California, Berkeley, California, 1981.

    Google Scholar 

  4. Stadler, W.,Applications of Multicriterion Optimization in Engineering and the Sciences, MCDM—Past Decade and Future Trends, Edited by M. Zeleny, JAI Press, Greenwich, Connecticut, pp. 52–84, 1984.

    Google Scholar 

  5. Stadler, W.,Applications of Multicriterion Optimization in Mechanics, Applied Mechanics Reviews, Vol. 37, pp. 277–286, 1984.

    Google Scholar 

  6. Stadler, W.,Sufficient Conditions for Preference Optimality, Journal of Optimization Theory and Applications, Vol. 18, pp. 119–140, 1976.

    Google Scholar 

  7. Stadler, W.,Preference Optimality in Multicriteria Control and Programming Problems, Journal of Nonlinear Analysis: Theory, Methods, and Applications, Vol. 4, pp. 51–65, 1979.

    Google Scholar 

  8. Vogel, W.,Vektoroptimierung in Produkträumen, Anton Hain Verlag, Meisenheim am Glan, Germany, 1977.

    Google Scholar 

  9. Vogel, W.,Optimization Theory and Nontransitive Preference Relations, Reprint No. 499, Sonderforschungsbereich 72, Institut für Angewandte Mathematik, Friedrich Wilhelm Universität, Bonn, Germany, 1982.

    Google Scholar 

  10. Borwein, J. M.,Proper Efficient Points for Maximizations with Respect to Cones, SIAM Journal on Control and Optimization, Vol. 15, pp. 57–63, 1977.

    Google Scholar 

  11. Borwein, J. M.,The Geometry of Pareto Efficiency over Cones, Mathematische Operationsforschung and Statistik, Serie Optimization, Vol. 11, pp. 235–248, 1980.

    Google Scholar 

  12. Jahn, J.,A Characterization of Properly Minimal Elements of a Set, SIAM Journal on Control and Optimization, Vol. 23, pp. 649–656, 1985.

    Google Scholar 

  13. Dauer, J. P., andSaleh, O.,Proper Efficiency with Respect to Cones, Preprint, University of Nebraska, Lincoln, Nebraska, 1986.

    Google Scholar 

  14. Geoffrion, A. M.,Proper Efficiency and the Theory of Vector Maximization, Journal of Mathematical Analysis and Applications, Vol. 22, pp. 618–630, 1968.

    Google Scholar 

  15. Kuhn, H. W., andTucker, A. W.,Nonlinear Programming, Proceeding of the Second Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, California, pp. 481–492, 1951.

  16. Minami, M.,Weak Pareto Optimality of Multiobjective Problem in a Banach Space, Bulletin of Mathematical Statistics, Vol. 19, pp. 19–23, 1981.

    Google Scholar 

  17. Minami, M.,Weak Pareto Optimality of Multiobjective Problems in a Locally Convex Linear Topological Space, Journal of Optimization Theory and Applications, Vol. 34, pp. 469–484, 1981; Errata, Vol. 38, pp. 149–150, 1982.

    Google Scholar 

  18. Minami, M.,Weak Pareto-Optimal Necessary Conditions in a Nondifferentiable Multiobjective Program on a Banach Space, Journal of Optimization Theory and Applications, Vol. 41, pp. 451–461, 1983.

    Google Scholar 

  19. Olech, C.,Existence Theorems for Optimal Problems with Vector-Valued Cost Function, Transactions of the American Mathematical Society, Vol. 136, pp. 159–180, 1967.

    Google Scholar 

  20. Gähler, S., Zur Polyoptimierung in Verallgemeinerten Topologischen Vektorräumen, Abhandlungen der Akademie der Wissenschaften der DDR, Abteilung Mathematik, Naturwissenschaft, Technik, Vol. 1, pp. 319–326, 1977.

    Google Scholar 

  21. Olech, C.,Existence Theorems for Optimal Control Problems Involving Multiple Integrals, Journal of Differential Equations, Vol. 6, pp. 512–526, 1969.

    Google Scholar 

  22. Cesari, L., andSuryanarayana, M. B.,Existence Theorems for Pareto Optimization; Multivalued and Banach Space Valued Functionals, Transactions of the American Mathematical Society, Vol. 244, pp. 37–65, 1978.

    Google Scholar 

  23. Corley, H. W.,An Existence Result for Maximization with Respect to Cones, Journal of Optimization Theory and Applications, Vol. 31, pp. 277–281, 1980.

    Google Scholar 

  24. Cesari, L., andSuryanarayana, M. B.,Existence Theorems for Pareto Problems of Optimization, Calculus of Variations and Control Theory, Edited by D. Russell, Academic Press, New York, New York, pp. 139–154, 1976.

    Google Scholar 

  25. Cesari, L., andSuryanarayana, M. B.,On Recent Existence Theorems in the Theory of Optimization, Journal of Optimization Theory and Applications, Vol. 31, pp. 397–415, 1980.

    Google Scholar 

  26. Suryanarayana, M. B.,Remarks on Existence Theorems for Pareto Optimality, Dynamical Systems, Proceedings of a University of Florida International Symposium, Edited by A. R. Bednarck and L. Cesari, Academic Press, New York, New York, pp. 335–347, 1977.

    Google Scholar 

  27. Cesari, L. andSuryanarayana, M. B.,An Existence Theorem for Pareto Problems, Nonlinear Analysis Theory, Methods, and Applications, Vol. 2, pp. 225–233, 1978.

    Google Scholar 

  28. Bacopoulos, A., andSinger, I.,On Convex Vectorial Optimization in Linear Spaces, Journal of Optimization Theory and Applications, Vol. 21, pp. 175–188, 1977; Errata, Vol. 23, pp. 473–476, 1977.

    Google Scholar 

  29. Malivert, C.,A Descent Method for Pareto Optimization, Journal of Mathematical Analysis and Applications, Vol. 88, pp. 610–631, 1982.

    Google Scholar 

  30. Breckner, W. W., andKolumbán, I., Dualität bei Optimierungsaufgaben in Topologischen Vektorräumen, Mathematica, Vol. 10, pp. 229–244, 1968.

    Google Scholar 

  31. Breckner, W. W., andKolumbán, I., Konvexe Optimierungsaufgaben in Topologischen Vektorräumen, Mathematica Scandinavica, Vol. 25, pp. 227–247, 1969.

    Google Scholar 

  32. Breckner, W. W., Dualität bei Optimierungsaufgaben in Halbgeordneten Topologischen Vektorräumen, Revue d'Analyse Numerique et de la Théorie de L'Approximation, Vol. 1, pp. 5–35, 1972.

    Google Scholar 

  33. Nieuwenhuis, J. W.,Supremal Points and Generalized Duality, Mathematische Operationsforschung and Statistik, Serie Optimization, Vol. 11, pp. 41–59, 1980.

    Google Scholar 

  34. Vogel, W.,Halbnormen and Vektoroptimierung, Quantitative Wirtschaftsforschung-Wilhelm Krelle zum 60. Geburtstag, Edited by H. Alback, E. Helmstädter, and R. Henn, Tübingen, Germany, pp. 703–714, 1977.

  35. Isac, G., Sur l'Existence de l'Optimum de Pareto, Rivista di Matematica dell'Università di Parma, Vol. 4, pp. 303–325, 1983.

    Google Scholar 

  36. Maschler, M., andPeleg, B.,Stable Sets and Stable Points of Set-Valued Dynamic Systems with Applications to Game Theory, SIAM Journal on Control and Optimization, Vol. 14, pp. 985–995, 1976.

    Google Scholar 

  37. Borwein, J. M.,On the Existence of Pareto Efficient Points, Mathematics of Operations Research, Vol. 8, pp. 54–73, 1983.

    Google Scholar 

  38. Hazen, G. B., andMorin, T. L.,Optimality Conditions in Nonconical Multiple-Objective Programming, Journal of Optimization Theory and Applications, Vol. 40, pp. 25–60, 1983.

    Google Scholar 

  39. Brown, T. A., andStrauch, R. E.,Dynamic Programming in Multiplicative Lattices, Journal of Mathematical Analysis and Applications, Vol. 12, pp. 364–370, 1965.

    Google Scholar 

  40. Bergstrom, T. C.,Maximal Elements of Acyclic Relations on Compact Sets, Journal of Economic Theory, Vol. 10, pp. 403–404, 1975.

    Google Scholar 

  41. Salukvadze, M.,An Approach to the Solution of the Vector Optimization Problem of Dynamic Systems, Journal of Optimization Theory and Applications, Vol. 38, pp. 409–422, 1982.

    Google Scholar 

  42. Wierzbicki, A.,Penalty Methods in Solving Optimization Problems with Vector Performance Criteria, Technical Report No. 12/1974, Institute of Automatic Control, Technical University of Warsaw, Warsaw, Poland, 1974.

    Google Scholar 

  43. Rolewicz, S.,On a Norm Scalarization in Infinite-Diumensional Banach Spaces, Control and Cybernetics, Vol. 4, pp. 85–89, 1975.

    Google Scholar 

  44. Penot, J. P., Calcul Sous-Differentiel et Optimisation, Journal of Functional Analysis, Vol. 27, pp. 248–276, 1978.

    Google Scholar 

  45. Athans, M., andGeering, H. P.,Necessary and Sufficient Conditions for Differentiable Nonscalar-Valued Functions to Attain Extrema, IEEE Transactions on Automatic Control, Vol. AC-18, pp. 132–138, 1973.

    Google Scholar 

  46. Isac, G., andIchim, I.,Sur un Critaire Suffisant d'Extremum dans des Espaces de Banach Ordonnés, Report No. 130, Institut de Mathematique Pure et Appliquée, Université Catholique de Louvain, Louvain-La-Neuve, France, 1979.

    Google Scholar 

  47. Ritter, K.,Optimization Theory in Linear Spaces, Part 3: Mathematical Programming in Partially Ordered Banach Spaces, Mathematische Annalen, Vol. 184, pp. 133–154, 1970.

    Google Scholar 

  48. Corley, H. W.,On Optimality Conditions for Maximizations with Respect to Cones, Journal of Optimization Theory and Applications, Vol. 46, pp. 67–78, 1985.

    Google Scholar 

  49. Bu, Q. Y., andShen, H. R.,Some Properties of Efficient Solutions for Vector Optimization, Journal of Optimization Theory and Applications, Vol. 46, pp. 255–263, 1985.

    Google Scholar 

  50. Vársen, C.,Sufficient Conditions of Optimality of Second Order in Banach Spaces, Revue Roumaine de Mathématiques Pures et Appliquées, Vol. 20, pp. 239–251, 1975.

    Google Scholar 

  51. Hoffman, K. H., andKornstaedt, H. J.,Higher-Order Necessary Conditions in Abstract Mathematical Programming, Journal of Optimization Theory and Applications, Vol. 26, pp. 533–568, 1978.

    Google Scholar 

  52. Slater, M.,Lagrange Multipliers Revisited: A Contribution to Nonlinear Programming, Cowles Commission Discussion Paper, Mathematics 403, 1950.

  53. Hurwicz, L., andUzawa, H.,A Note on the Lagrangian Saddle-Points, Studies in Linear and Nonlinear Programming, Edited by K. J. Arrow, L. Hurwicz, and H. Uzawa, Stanford University Press, Stanford, California, pp. 103–113, 1958.

    Google Scholar 

  54. Klee, V. L., Jr.,Separation Properties of Convex Cones, Proceedings of the American Mathematical Society, Vol. 6, pp. 313–318, 1955.

    Google Scholar 

  55. Krein, M. G., andRutman, M. A.,Linear Operators Leaving Invariant a Cone in a Banach Space, Uspekhi Matematicheskie Nauk (NS), Vol. 3, pp. 3–95, 1948; American Mathematical Society Translation No. 26, New York, New York, 1950.

    Google Scholar 

  56. DaCunha, N. O., andPolak, E.,Constrained Minimization under Vector-Valued Criteria in Linear Topological Spaces, Mathematical Theory of Control, Edited by A. V. Balakrishnan and L. W. Neustadt, Academic Press, New York, New York, pp. 96–108, 1967.

    Google Scholar 

  57. Majumdar, M.,Some Approximation Theorems on Efficiency Prices for Infinite Programs, Journal of Economic Theory, Vol. 2, pp. 399–410, 1970.

    Google Scholar 

  58. Majumdar, M.,Efficient Programs in Infinite-Dimensional Spaces: A Complete Characterization, Journal of Economic Theory, Vol. 7, pp. 355–369, 1974.

    Google Scholar 

  59. Radner, R.,A Note on Maximal Points of Convex Sets in l ∞, Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, California, pp. 351–354, 1967.

    Google Scholar 

  60. Radner, R.,Efficiency Prices for Infinite-Horizon Production Programmes, Review of Economic Studies, Vol. 34, pp. 51–66, 1967.

    Google Scholar 

  61. Majumdar, M.,Some General Theorems on Efficiency Prices with an Infinite-Dimensional Commodity Space, Journal of Economic Theory, Vol. 5, pp. 1–13, 1972.

    Google Scholar 

  62. Peleg, B., andYaari, M. E.,Efficiency Prices in an Infinite-Dimensional Space, Journal of Economic Theory, Vol. 2, pp. 41–85, 1970.

    Google Scholar 

  63. Jahn, J.,Scalarization in Vector Optimization, Mathematical Programming, Vol. 29, pp. 203–218, 1984.

    Google Scholar 

  64. Simon, C. P.,Conditions for Constrained Pareto Optima on a Banach Space with a Finite Number of Criteria, Dynamical Systems, Proceedings of a University of Florida International Symposium, Edited by A. R. Bednarck and L. Cesari, Academic Press, New York, New York, pp. 323–334, 1977.

    Google Scholar 

  65. Bacopoulos, A., Godini, G., andSinger, I.,Infima of Sets in the Plane and Applications to Vectorial Optimization, Revue Roumaine de Mathématiques Pures et Appliquées, Vol. 23, pp. 343–360, 1978.

    Google Scholar 

  66. Gearhart, W. B.,On the Characterization of Pareto-Optimal Solutions in Bicriterion Optimization, Journal of Optimization Theory and Applications, Vol. 27, pp. 301–307, 1979.

    Google Scholar 

  67. Lantos, B.,Necessary Conditions for the Optimality in Abstract Optimum Control Problems with Nonscalar-Valued Performance Criterion, Problems of Control and Information Theory, Vol. 5, pp. 271–284, 1976.

    Google Scholar 

  68. Neustadt, L. W.,A General Theory of Extremals, Journal of Computer and System Sciences, Vol. 3, pp. 57–92, 1969.

    Google Scholar 

  69. Azimov, A. Ja.,Smooth-Convex Vector Optimization Problems, Soviet Mathematics Doklady, Vol. 23, pp. 154–158, 1981.

    Google Scholar 

  70. Kutateladze, S. S.,Convex ɛ-Programming, Soviet Mathematics Doklady, Vol. 20, pp. 391–393, 1979.

    Google Scholar 

  71. Luenberger, D. G.,Optimization by Vector Space Methods, John Wiley and Sons, New York, New York, 1969.

    Google Scholar 

  72. Tang, Y.,Conditions for Constrained Efficient Solutions of Multiobjective Problems in Banach Spaces, Journal of Mathematical Analysis and Applications, Vol. 96, pp. 505–519, 1983.

    Google Scholar 

  73. Fenchel, W.,Convex Cones, Sets, and Functions, Lecture Notes, Department of Mathematics, Princeton University, Princeton, New Jersey, 1953.

    Google Scholar 

  74. Rockafellar, R. T.,Extension of Fenchel's Duality Theorem for Convex Functions, Duke Mathematics Journal, Vol. 33, pp. 81–89, 1966.

    Google Scholar 

  75. Zowe, J.,A Duality Theorem for a Convex Programming Problem in Order-Complete Vector Lattices, Journal of Mathematical Analysis and Applications, Vol. 50, pp. 273–287, 1975.

    Google Scholar 

  76. Azimov, A. Ja.,Duality in Vector Optimization Problems, Soviet Mathematics Doklady, Vol. 26, pp. 170–174, 1982.

    Google Scholar 

  77. Rosinger, E. E.,Duality and Alternative in Multiobjective Optimization, Proceedings of the American Mathematical Society, Vol. 64, pp. 307–312, 1977.

    Google Scholar 

  78. Rosinger, E. E.,Multiobjective Duality without Convexity, Journal of Mathematical Analysis and Applications, Vol. 66, pp. 442–450, 1978.

    Google Scholar 

  79. Thera, M.,Subdifferential Calculus for Convex Operators, Journal of Mathematical Analysis and Applications, Vol. 80, pp. 78–91, 1981.

    Google Scholar 

  80. Lehmann, R., andOettli, W.,The Theorem of the Alternative, the Key Theorem, and the Vector-Maximum Problem, Mathematical Programming, Vol. 8, pp. 332–344, 1975.

    Google Scholar 

  81. Oettli, W.,A Duality Theorem for the Nonlinear Vector Maximum Problem, Colloquia Mathematica Societatis János Bolyai, Vol. 12, Progress in Operations Research, Edited by A. Prekopa, Eger, Hungary, 1974; North-Holland, Amsterdam, Holland, 1976.

  82. Jahn, J.,Duality in Vector Optimization, Mathematical Programming Vol. 25, pp. 343–353, 1983.

    Google Scholar 

  83. Gearhart, W. B.,On Vectorial Approximation, Journal of Approximation Theory, Vol. 10, pp. 49–63, 1974.

    Google Scholar 

  84. Jahn, J.,Scalarization in Multi-Objective Optimization, Preprint No. 847, Fachbereich Mathematik, Technische Hochschule Darmstadt, 1984.

  85. Cesari, L., andSuryanarayana, M. B.,Existence Theorems for Pareto Optimization in Banach Spaces, Bulletin of the American Mathematical Society, Vol. 82, pp. 306–308, 1976.

    Google Scholar 

  86. Chang, S. S. L.,General Theory of Optimal Processes, SIAM Journal on Control, Vol. 4, pp. 46–55, 1966.

    Google Scholar 

  87. Geering, H. P., andAthans, M.,The Infimum Principle, IEEE Transactions on Automatic Control, Vol. AC-19, pp. 485–494, 1974.

    Google Scholar 

  88. Leitmann, G.,Some Problems of Scalar and Vector-Valued Optimization in Linear Viscoelasticity, Journal of Optimization Theory and Applications, Vol. 23, pp. 93–99, 1977.

    Google Scholar 

  89. Leitmann, G., andSchmitendorf, W.,Some Sufficiency Conditions for Pareto-Optimal Control, Journal of Dynamical Systems, Measurement, and Control, Vol. 95, pp. 356–361, 1973.

    Google Scholar 

  90. Lukka, M.,An Algorithm for Solving a Multiple-Criteria Optimal Control Problem, Journal of Optimization Theory and Applications, Vol. 28, pp. 435–438, 1979.

    Google Scholar 

  91. Rubio, J. E.,Solution of Nonlinear Optimal Control Problems in Hilbert Spaces by Means of Linear Programming Techniques, Journal of Optimization Theory and Applications, Vol. 30, pp. 643–661, 1980.

    Google Scholar 

  92. Sakawa, M.,An Approximate Solution of Linear Multicriteria Control Problems through the Multicriteria Simplex Method, Journal of Optimization Theory and Applications, Vol. 22, pp. 417–427, 1977.

    Google Scholar 

  93. Sakawa, M.,Solution of Multicriteria Control Problems in Certain Types of Linear Distributed-Parameter Systems by a Multicriteria Simplex Method, Journal of Mathematical Analysis and Applications, Vol. 64, pp. 181–188, 1978.

    Google Scholar 

  94. Salukavadze, M. E., Zadachi Vektorni Optimizatsii v Teorii Upravlenia, Metsniereba Publishing House, Moscow, USSR, 1975; translated into English by J. L. Casti asVector Valued Optimization Problems in Control Theory, Academic Press, New York, New York, 1979.

    Google Scholar 

  95. Schmitendorf, W. E.,Optimal Control of Systems with Multiple Criteria when Disturbances Are Present, Journal of Optimization Theory and Applications, Vol. 27, pp. 135–146, 1979.

    Google Scholar 

  96. Schmitendorf, W. E., andLeitmann, G.,A Simple Derivation of Necessary Conditions for Pareto Optimality, IEEE Transactions on Automatic Control, Vol. AC-19, pp. 601–602, 1974.

    Google Scholar 

  97. Van Slyke, R. M., andWets, R. J. B.,A Duality Theory for Abstract Mathematical Programs with Applications to Optimal Control Theory, Journal of Mathematical Analysis and Applications, Vol. 22, pp. 679–706, 1968.

    Google Scholar 

  98. Stadler, W., andLeitmann, G.,Cooperative Games for the Experimentalist, Zagadnienia Drgan Nieliniowych [Journal of Nonlinear Vibration Problems], Warsaw, Poland, Vol. 18, pp. 273–286, 1974; Proceedings, 6th International Conference on Nonlinear Oscillations, Poznan, Poland, 1972.

    Google Scholar 

  99. Stadler, W.,Natural Structural Shapes), Quarterly Journal of Mechanics and Applied Mathematics, Vol. 31, pp. 169–217, 1978.

    Google Scholar 

  100. Stadler, W.,Stability of the Natural Shapes of Sinusoidally Loaded Uniform Shallow Arches, Quarterly Journal of Mechanics and Applied Mathematics, Vol. 36, pp. 365–386, 1983.

    Google Scholar 

Supplementary Bibliography (Books)

  1. Schaeffer, H. H.,Topological Vector Spaces, Springer Verlag, New York, New York, 1971.

    Google Scholar 

  2. Peressini, A. L.,Ordered Topological Vector Spaces, Harper and Row, New York, New York, 1967.

    Google Scholar 

  3. Jameson, G.,Ordered Linear Spaces, Springer Verlag, Berlin, Germany, 1979.

    Google Scholar 

  4. Köthe, G.,Topologische Lineare Räume, I., Springer Verlag, Berlin, Germany, 1960.

    Google Scholar 

  5. Robertson, A. P., andRobertson, W.,Topological Vector Spaces, 2nd Edition, Cambridge University Press, Cambridge, England, 1973.

    Google Scholar 

Supplementary Bibliography (Articles)

  1. Athans, M., andGeering, H. P.,Necessary and Sufficient Conditions for Differentiable Nonscalar-Valued Functions to Attain Extrema, IEEE Transactions on Automatic Control, Vol. AC-18, pp. 132–138, 1973.

    Google Scholar 

  2. Azimov, A. Ja.,Smooth-Convex Vector Optimization Problems, Soviet Mathematics Doklady, Vol. 23, pp. 154–158, 1981.

    Google Scholar 

  3. Azimov, A. Ja.,Duality in Vector Optimization Problems, Soviet Mathematics Doklady, Vol. 26, pp. 170–174, 1982.

    Google Scholar 

  4. Borwein, J. M., andNieuwenhuis, J. W.,Two Kinds of Normality in Vector Optimization, Mathematical Programming, Vol. 28, pp. 185–191, 1984.

    Google Scholar 

  5. Elster, K. H., andNehse, R.,Necessary and Sufficient Conditions for the Order-Completeness of Partially Ordered Vector Spaces, Mathematische Nachrichten, Vol. 81, pp. 301–311, 1978.

    Google Scholar 

  6. Gähler, S., Zur Polyoptimierung in Verallgemeinerten Topologischen Vektorräumen, Akademie der Wissenschaften DDR, Abteilung Mathematik, Naturwissenschaft, Tecknik, Vol. 1, pp. 319–326, 1977.

    Google Scholar 

  7. Geering, H. P., andAthans, M.,The Infimum Principle, IEEE Transactions on Automatic Control, Vol. AC-19, pp. 485–494, 1974.

    Google Scholar 

  8. Isac, G., andIchim, I.,Sur un Critaire Suffisant d'Extremum dans des Espaces de Banach Ordonnés, Report No. 130, Institut de Mathematique Pure et Appliquée, Université Catholique de Louvain, Louvain-La-Neuve, France, 1979.

    Google Scholar 

  9. Kusraev, A. G.,On Necessary Conditions for an Extremum of Nonsmooth Vector-Valued Mappings, Soviet Mathematics Doklady, Vol. 19, pp. 1057–1060, 1978.

    Google Scholar 

  10. Kutateladze, S. S.,Extreme Points of Subdifferentials, Soviet Mathematics Doklady, Vol. 19, pp. 1235–1237, 1978.

    Google Scholar 

  11. Kutateladze, S. S.,Convex ∈-Programming, Soviet Mathematics Doklady, Vol. 20, pp. 391–393, 1979.

    Google Scholar 

  12. Kutateladze, S. S., andFel'dman, M. M.,Lagrange Multipliers in Vector Optimization Problems, Soviet Mathematics Doklady, Vol. 17, pp. 1510–1514, 1976.

    Google Scholar 

  13. Lantos, B.,Necessary Conditions for Optimality in Abstract Optimum Control Problems with Nonscalar-Valued Performance Criterion, Problems of Control and Information Theory, Vol. 5, pp. 271–284, 1976.

    Google Scholar 

  14. Lata, M.,Strong Pseudo-Convex Programming in Banach Space, Indian Journal of Pure and Applied Mathematics, Vol. 6, pp. 45–48, 1976.

    Google Scholar 

  15. Nieuwenhuis, J. W.,Supremal Points and Generalized Duality, Mathematische Operationsforschung and Statistik, Serie Optimization, Vol. 11, pp. 41–59, 1980.

    Google Scholar 

  16. Penot, J. P., Calcul Sous-Differentiel et Optimisation, Journal of Functional Analysis, Vol. 27, pp. 248–276, 1978.

    Google Scholar 

  17. Ritter, K.,Optimization Theory in Linear Spaces, Part 1, Mathematische Annalen, Vol. 182, pp. 189–206, 1969.

    Google Scholar 

  18. Ritter, K.,Optimization Theory in Linear Spaces, Part 2: On Systems of Linear Operator Inequalities in Partially-Ordered Normed Linear Spaces, Mathematische Annalen, Vol. 183, pp. 169–180, 1969.

    Google Scholar 

  19. Ritter, K.,Optimization Theory in Linear Spaces, Part 3: Mathematical Programming in Partially-Ordered Banach Spaces, Mathematische Annalen, Vol. 184, pp. 133–154, 1970.

    Google Scholar 

  20. Rolewicz, S.,On a Norm Scalarization in Infinite-Dimensional Banach Spaces, Control and Cybernetics, Vol. 4, pp. 85–89, 1975.

    Google Scholar 

  21. Rosinger, E. E.,Duality and Alternative in Multiobjective Optimization, Proceedings of the American Mathematical Society, Vol. 64, pp. 307–312, 1977.

    Google Scholar 

  22. Rosinger, E. E.,Multiobjective Duality without Convexity, Journal of Mathematical Analysis and Applications, Vol. 66, 442–450, 1978.

    Google Scholar 

  23. Rubinov, A. M.,Sublinear Operators and Their Applications, Russian Mathematical Surveys, Vol. 32, pp. 115–175, 1977.

    Google Scholar 

  24. Thera, M.,Subdifferential Calculus for Convex Operators, Journal of Mathematical Analysis and Applications, Vol. 80, pp. 78–91, 1981.

    Google Scholar 

  25. Thibault, L.,Subdifferentials of Compactly Lipschitzian Vector-Valued Functions, Annali di Matematica Pura ed Applicata, Vol. 125, pp. 157–192, 1980.

    Google Scholar 

  26. Vogel, W.,Vektoroptimierung in Produkträumen, Anton Hain Verlag, Meisenheim am Glan, Germany, 1977.

    Google Scholar 

  27. Wierzbicki, A.,Penalty Methods in Solving Optimization Problems with Vector Performance Criteria, Technical Report No. 12/1974, Institute of Automatic Control, Technical University of Warsaw, Warsaw, Poland, 1974.

    Google Scholar 

  28. Wierzbicki, A. P., andKurcyusz, S.,Projection on a Cone, Penalty Functionals and Duality Theory for Problems with Inequality Constraints in Hilbert Space, SIAM Journal on Control and Optimization, Vol. 15, pp. 25–56, 1977.

    Google Scholar 

  29. Zowe, J.,A Duality Theorem for a Convex Programming Problem in Order-Complete Vector Lattices, Journal of Mathematical Analysis and Applications, Vol. 50, pp. 273–287, 1975.

    Google Scholar 

  30. Zowe, J.,The Saddle Point Theorem of Kuhn and Tucker in Ordered Vector Spaces, Journal of Mathematical Analysis and Applications, Vol. 57, pp. 41–55, 1977.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Nebraska, Lincoln, Nebraska

    J. P. Dauer (Professor)

  2. Division of Engineering, San Francisco State University, San Francisco, California

    W. Stadler (Professor)

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The authors wish to express their appreciation to O. Saleh and Y. H. Liu for their thoughtful ear, their cogent suggestions, and their untiring legwork in procuring and discussing many of the papers cited in this review. Above all, however, their thanks go to Professors L. Hurwicz and J. M. Borwein for the excellent papers they produced in this area. Their work was a pleasure to read, and it provided the supporting framework without which our task would have been a considerably more difficult one.

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Dauer, J.P., Stadler, W. A survey of vector optimization in infinite-dimensional spaces, part 2. J Optim Theory Appl 51, 205–241 (1986). https://doi.org/10.1007/BF00939823

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