Abstract
In this paper, we are concerned with mathematical programs which construct nonmajorizing vectors for several specific majorization applications. In particular, we develop a linear integer program and two integer goal programs which solve the assignment majorization problem. We also develop a quadratic program for solving majorization problems which arise in probability and statistics. In the appendix, we present a general goal-programming algorithm for these, as well as others, goal programs.
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Hardy, G. H., Littlewood, J. E., andPólya, G.,Inequalities, Cambridge University Press, Cambridge, England, 1952.
Marshall, A. W., andProschan, F.,An Inequality for Convex Functions Involving Majorization, Journal of Mathematical Analysis and Applications, Vol. 12, pp. 87–90, 1965.
Ostrowski, P. A.,Sur Quelques Applications des Fonctions Convexes et Concaves au Sens de L. Schur, Journal de Mathématiques Pures et Appliquées, Vol. 31, pp. 253–292, 1952.
Haimes, Y. Y., Hall, W. A., andFreedman, H. T.,Multiobjective Optimization in Water Resources Systems, Elsevier Scientific Publishing, New York, New York, 1975.
Zadeh, L. A.,Optimality and Non-Scalar-Valued Performance Criteria, IEEE Transactions on Automatic Control, Vol. AC-8, pp. 59–60, 1963.
Olkin, I.,Monotonicity Properties of Dirichlet Integrals with Applications to the Multinomial Distribution and the Analysis of Variance, Biometrika, Vol. 59, pp. 303–307, 1972.
Rinott, Y.,Multivariate Majorization and Rearrangement Inequalities with Some Applications to Probability and Statistics, Israel Journal of Mathematics, Vol. 15, pp. 60–77, 1973.
Pledger, G., andProschan, F.,Comparisons of Order Statistics and of Spacings from Heterogeneous Distributions, Optimizing Methods in Statistics, Edited by J. S. Rustage, Academic Press, New York, New York, pp. 89–113, 1971.
Proschan, F.,Applications of Majorization and Schur Functions in Reliability and Life Testing, Reliability and Fault Tree Analysis, Edited by R. E. Barlow, J. B. Fussell, and N. D. Singpurwalla, SIAM, Philadelphia, Pennsylvania, 1975.
Proschan, F., andSethuraman, J.,Stochastic Comparisons of Order Statistics from Heterogeneous Populations, with Applications in Reliability, Journal of Multivariate Analysis, Vol. 6, pp. 608–616, 1976.
Tamir, A.,Polyhedral Sets Containing a Least Majorized Element, Bulletin of the Institute of Mathematical Statistics, Vol. 34, p. 200, 1977.
DaCunha, N. O., andPolak, E.,Constrained Minimization under Vector-Valued Criteria in Finite-Dimensional Spaces, Journal of Mathematical Analysis and Applications, Vol. 19, pp. 103–124, 1967.
Charnes, A., andCooper, W. W.,Management Models and Industrial Applications of Linear Programming, Vol. 1, John Wiley and Sons, New York, New York, 1961.
Dauer, J. P., andKrueger, R. J.,An Iterative Approach to Goal Programming, Operational Research Quarterly, Vol. 28, pp. 671–681, 1977.
Lee, S. M.,Goal Programming for Decision Analysis, Auerbach Publishers, Philadelphia, Pennsylvania, 1972.
Lee, S. M.,Interactive and Integer Goal Programming, Management Science (to appear).
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Communicated by G. Leitmann
The authors would like to thank Professor Y. Tong for introducing them to majorization and its applications. They would also like to thank Professors Y. Tong and D. Mesner for helpful discussions.
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Dauer, J.P., Krueger, R.J. Mathematical programming techniques in majorization. J Optim Theory Appl 25, 361–373 (1978). https://doi.org/10.1007/BF00932899
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DOI: https://doi.org/10.1007/BF00932899