×

The structure of admissible points with respect to cone dominance. (English) Zbl 0389.52021


MSC:

52A40 Inequalities and extremum problems involving convexity in convex geometry
90C31 Sensitivity, stability, parametric optimization
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Debreu, G.,Representation of a Preference Ordering by a Numerical Function, Decision Processes, Edited by R. M. Thrall, C. H. Coombs, and R. L. Davis, John Wiley and Sons, New York, New York, 1954. · Zbl 0058.13803
[2] Debreu, G.,Theory of Value, John Wiley and Sons, New York, New York, 1959. · Zbl 0193.20205
[3] Bowen, R.,A New Proof of a Theorem in Utility Theory, International Economic Review, Vol. 9, p. 374, 1968. · Zbl 0164.50307 · doi:10.2307/2556234
[4] Arrow, K. J., andHahn, F. H.,General Competitive Equilibrium, Holden-Day, San Francisco, California, 1971. · Zbl 0311.90001
[5] Keeney, R., andRaiffa, H.,Decisions with Multiple Objectives: Preferences and Value Trade-offs, John Wiley and Sons, New York, New York, 1976.
[6] MacCrimmon, K. R.,An Overview of Multiple-Objective Decision-Making, Multiple Criteria Decision Making, Edited by J. L. Cochrane and M. Zeleny, University of South Carolina Press, Columbus, South Carolina, 1973.
[7] Wald, A.,Statistical Design Functions, John Wiley and Sons, New York, New York, 1950. · Zbl 0040.36402
[8] Arrow, K. J., Barankin, E. W., andBlackwell, D.,Admissible Points of Convex Sets, Contributions to the Theory of Games, Vol. 2, Edited by H. W. Kuhn and A. W. Tucker, Princeton University Press, Princeton, New Jersey, 1953. · Zbl 0050.14203
[9] Gale, D., Kuhn, H. W., andTucker, A. W.,Linear Programming and the Theory of Games, Activity Analysis of Production and Allocation, Edited by T. C. Koopmans, John Wiley and Sons, New York, New York, 1951.
[10] Kuhn, H. W., andTucker, A. W.,Nonlinear Programming, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, California, 1950.
[11] Koopmans, T. C.,Three Essays on the State of Economic Science, McGraw-Hill Book Company, New York, New York, 1957. · Zbl 0080.34901
[12] Geoffrion, A. M.,Proper Efficiency and the Theory of Vector Maximization, Journal of Mathematical Analysis and Applications, Vol. 22, pp. 618-630, 1968. · Zbl 0181.22806 · doi:10.1016/0022-247X(68)90201-1
[13] Geoffrion, A. M.,Strictly Concave Parametric Programming, Parts I and II, Management Science, Vol. 13, pp. 244-253 and 359-370, 1967. · Zbl 0143.42604 · doi:10.1287/mnsc.13.3.244
[14] Geoffrion, A. M.,Solving Bicriterion Mathematical Programs, Operations Research, Vol. 15, pp. 39-54, 1967. · Zbl 0173.21602 · doi:10.1287/opre.15.1.39
[15] Yu, P. L.,Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Decision Problems with Multiobjectives, Journal of Optimization Theory and Applications, Vol. 14, pp. 319-377, 1974. · Zbl 0268.90057 · doi:10.1007/BF00932614
[16] Smale, S.,Global Analysis and Economics, III: Pareto Optima and Price Equilibria, Journal of Mathematical Economics, Vol. 1, pp. 107-117, 1974. · Zbl 0316.90007 · doi:10.1016/0304-4068(74)90002-0
[17] Smale, S.,Global Analysis and Economics, V: Pareto Theory with Constraints, Journal of Mathematical Economics, Vol. 1, pp. 213-221, 1974. · Zbl 0357.90010 · doi:10.1016/0304-4068(74)90013-5
[18] Smale, S.,Global Analysis and Economics, VI: Geometric Analysis of Pareto Optima and Price Equilibria under Classical Hypotheses, Journal of Mathematical Economics, Vol. 3, pp. 1-14, 1976. · Zbl 0348.90017 · doi:10.1016/0304-4068(76)90002-1
[19] Rand, D.,Thresholds in Pareto Sets, Journal of Mathematical Economics, Vol. 3, pp. 139-154, 1976. · Zbl 0357.90016 · doi:10.1016/0304-4068(76)90023-9
[20] Simon, C. P., andTitus, C.,Characterization of Optima in Smooth Pareto Economic Systems, Journal of Mathematical Economics, Vol. 2, pp. 297-330, 1975. · Zbl 0348.90020 · doi:10.1016/0304-4068(75)90029-4
[21] Wan, Y. H.,On Local Pareto Optimum, Journal of Mathematical Economics, Vol. 2, pp. 35-42, 1975. · Zbl 0309.90049 · doi:10.1016/0304-4068(75)90012-9
[22] Charnes, A., andCooper, W. W.,Management Models and Industrial Applications of Linear Programming, Vol. 1, Chapter 9, John Wiley and Sons, New York, New York, 1961. · Zbl 0107.37004
[23] Ecker, J. G., andKouada, I. A.,Finding Efficient Points for Linear Multiple Objective Programs, Mathematical Programming, Vol. 8, pp. 375-377, 1975. · doi:10.1007/BF01580453
[24] Ecker, J. G., andKouada, I. A.,Generating Maximal Efficient Faces for Multiple Objective Linear Programs, Université Catholique de Louvain, Heverlee, Belgium, CORE, Discussion Paper No. 7617, 1976.
[25] Evans, J. P., andSteuer, R. E.,A Revised Simplex Method for Linear Multiple Objective Programs, Mathematical Programming, Vol. 5, pp. 54-72, 1973. · Zbl 0281.90045 · doi:10.1007/BF01580111
[26] Gal, T.,A General Method for Determining the Set of All Efficient Solutions to a Linear Vector Maximum Problem, Institut für Wirstschaftswissenschaften, Aachen, Germany, Report No. 76/12, 1976.
[27] Geoffrion, A. M., Dyer, J. S., andFeinberg, A.,An Interactive Approach for Multi-Criterion Optimization with an Application to the Operation of an Academic Department, Management Science, Vol. 19, pp. 357-368, 1972. · Zbl 0247.90069 · doi:10.1287/mnsc.19.4.357
[28] Philip, J.,Algorithms for the Vector Maximization Problem, Mathematical Programming, Vol. 2, pp. 207-229, 1972. · Zbl 0288.90052 · doi:10.1007/BF01584543
[29] Schachtman, R.,Generation of the Admissible Boundary of a Convex Polytope, Operations Research, Vol. 22, pp. 151-159, 1974. · Zbl 0277.52006 · doi:10.1287/opre.22.1.151
[30] Yu, P. L., andZeleny, M.,The Set of All Nondominated Solutions in Linear Cases and a Multicriteria Simplex Method, Journal of Mathematical Analysis and Applications, Vol. 49, pp. 430-468, 1975. · Zbl 0313.65047 · doi:10.1016/0022-247X(75)90189-4
[31] Ferguson, T. S.,Mathematical Statistics, Academic Press, New York, New York, 1967. · Zbl 0153.47602
[32] Whittle, P.,Optimization Under Constraints, Chapter 10, John Wiley and Sons, New York, New York, 1971. · Zbl 0218.90041
[33] Markowitz, H.,The Optimization of Quadratic Functions Subject to Linear Constraints, Naval Research Logistics Quarterly, Vol. 3, pp. 111-133, 1956. · doi:10.1002/nav.3800030110
[34] Markowitz, H.,Portfolio Selection: Efficient Diversification of Investments, John Wiley and Sons, New York, New York, 1962.
[35] Raiffa, H.,Decision Analysis: Introductory Lectures on Choice Under Uncertainty, Addison-Wesley Publishing Company, Reading, Massachusetts, 1970. · Zbl 0181.21802
[36] Bitran, G. R.,Admissible Points and Vector Optimization: A Unified Approach, Massachusetts Institute of Technology, Operations Research Center, PhD Thesis, 1975.
[37] Cochrane, J. L., andZeleny, M., Editors,Multiple Criteria Decision Making, University of South Carolina Press, Columbus, South Carolina, 1973.
[38] Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970. · Zbl 0193.18401
[39] Stoer, J., andWitzgall, C.,Convexity and Optimization in Finite Dimensions I, Springer-Verlag, Berlin, Germany, 1970. · Zbl 0203.52203
[40] Berge, C.,Topological Spaces, The MacMillan Company, New York, New York, 1963. · Zbl 0114.38602
[41] Hildenbrand, W., andKirman, A. P.,Introduction to Equilibrium Analysis, North-Holland Publishing Company, Amsterdam, Holland, 1976. · Zbl 0345.90004
[42] Mangasarian, O. L.,Nonlinear Programming, McGraw-Hill Book Company, New York, New York, 1969.
[43] Fiacco, A. V., andMcCormick, G. P.,Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley and Sons, New York, New York, 1968. · Zbl 0193.18805
[44] Magnanti, T. L.,A Linear Approximation Approach to Duality in Nonlinear Programming, Massachusetts Institute of Technology, Operations Research Center, Working Paper No. OR-016-73, 1973.
[45] Halkin, H.,Nonlinear Nonconvex Programming in an Infinite Dimensional Space, Mathematical Theory of Control Edited by A. V. Balakrishnan and L. W. Neustadt, Academic Press, New York, New York, 1967. · Zbl 0223.90032
[46] Robinson, S. M.,First Order Conditions for General Nonlinear Optimization, SIAM Journal on Applied Mathematics, Vol. 30, pp. 597-607, 1976. · Zbl 0364.90093 · doi:10.1137/0130053
[47] Borwein, J.,Proper Efficient Points for Maximizations with Respect to Cones, SIAM Journal on Control and Optimization, Vol. 15, pp. 57-63, 1977. · Zbl 0369.90096 · doi:10.1137/0315004
[48] Naccache, P.,Stability in Multicriteria Optimization, University of California at Berkeley, Department of Mathematics, PhD Thesis, 1977. · Zbl 0418.90079
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.