Abstract
This chapter discusses extensions of the methods developed in previous chapters and discusses some closely related subjects, including
-
(i)
Entropy optimization problems with a finite number of constraints but a countably infinite number of variables,
-
(ii)
A relationship between entropy optimization and Bayesian statistical estimation,
-
(iii)
Entropic regularization method for solving min-max problems, and
-
(iv)
Entropic regularization method for solving semi-infinite min-max problems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Apostol, A.M., Mathematical Analysis, Addison-Wesley Publishing, Redwood City, CA, 1977.
Ben-Tal, A., and Charnes, A., “A Dual Optimization Framework for Some Problems of Information Theory and Statistics,” Problems of Control and Information Theory, Vol. 8, 1979, pp. 387–401.
Ben-Tal, A., and Teboulle, M., “A Smooth Technique for Nondifferentiable Optimization Problems,” Optimization — Fifth French-German Conference, Castel Novel, 1988, Lecture Notes in Mathematics 1405, Springer Verlag, 1989, pp. 1-11.
Ben-Tal, A., Teboulle, M., and Charnes, A., “The Role of Duality in Optimization Problems involving Entropy Functionals with Applications to Information Theory,” Journal of Optimization Theory and Applications, Vol. 58, 1988, pp. 209–223.
Bertsekas, D., Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, 1989.
Billingsley, P., Probability and Measure, John Wiley, New York, 1979.
van Campenhout, J. and Cover, T.M., “Maximum Entropy and Conditional Probability,” IEEE Transactions on Information Theory, Vol. 27, 1981, pp. 483–489.
Charalambous, C., and Conn, A.R., “An Efficient Method to Solve the Minimax Problem Directly,” SIAM Journal on Numerical Analysis, Vol. 15, 1978, pp. 162–187.
Charnes, A., Cooper, W.W., and Seiford, L., “Extremal Principles and Optimization Qualities for Khinchin-Kullback-Leibler Estimation,” Mathematische Operationsforschung und Statistik, Series Optimization, Vol. 9, 1978, pp. 21–29.
DeGroot, M.H., Optimal Statistical Decisions, McGraw-Hill, New York, 1970.
Di Pillo, G., Grippo, L., and Lucidi, S., “A Smooth Method for the Finite Minimax Problem,” Mathematical Programming, Vol. 60, 1993, pp. 187–214.
Fang, S.-C., and Rajasekera, J.R., “Quadratically Constrained Minimum Cross-Entropy Analysis,” Mathematical Programming, Vol. 44, 1989, pp. 85–96.
Fang, S.-C., and Puthenpura, S., Linear Optimization and Extensions: Theory and Algorithms, Prentice Hall, Englewood Cliffs, New Jersey, 1993.
Fang, S.-C., and Wu, S.-Y., “Solving Min-Max Problems and Linear Semi-Infinite Programs,” Computers and Mathematics with Applications, Vol. 32, 1996, pp. 87–93.
Fletcher, R., Practical Methods of Optimization, Vol. 2, John Wiley, New York, 1981.
Gigola, C., and Gomez, S., “A Regularization Method for Solving the Finite Convex Min-Max Problem,” SIAM Journal on Numerical Analysis, Vol. 27, 1990, pp. 1621–1634.
Hettich, R., and Kortanek, K.O., “Semi-Infinite Programming: Theory, Method and Applications,” SIAM Review, Vol. 35, 1993, pp. 380–429.
Huard, P., “Resolution of Mathematical Programming with Nonlinear Constraints by the Method of Centers,” in Nonlinear Programming, edited by J. Abadie, North-Holland, Amsterdam, 1967, pp. 207–219.
Hiriart-Urruty, J.-B., and Lemarechal, C., Convex Analysis and Minimization Algorithm, Springer-Verlag, Berlin, 1993.
Li, X.S., and Fang, S.-C., “On the Entropic Regularization Method for Solving Min-Max Problems with Applications,” to appear in Mathematical Methods of Operations Research, Vol. 46, 1997.
Jaynes, E.T., “Information Theory and Statistical Mechanics,” Physics Review, Vol. 106, 1957, pp. 620–630.
Jaynes, E.T., “Information Theory and Statistical Mechanics II,” Physics Review, Vol. 108, 1957, pp. 171–190.
Jaynes, E.T., “Prior Probabilities,” IEEE Transactions. Systems Science and Cybernetics, Vol. 4, 1968, pp. 227–241.
Jaynes, E.T., “The Relation of Bayesian and Maximum Entropy Methods,” Maximum-Entropy and Bayesian Methods in Science and Engineering, Volume 1: Foundations, edited by G. J. Erickson and C. R. Smith, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988, pp. 25–29.
Kazarinoff, N.D., Analytic Inequalities, Holt, Rinehart and Winston, New York, 1961.
Kullback, S., Information and Statistics, John Wiley, New York, 1959.
Li, X.-S., Entropy and Optimization, Ph.D. Thesis, University of Liverpool, United Kingdom, 1987.
Lin, C.J., Fang, S.-C., and Wu, S.Y., Parametric Linear Semi-Infinite Programming, Applied Mathematics Letters, Vol. 9, 1996, pp. 89–96.
Lin, C.J., Fang, S.-C., and Wu, S.Y., An Unconstrained Convex Programming Approach to Linear Semi-Infinite Programming, OR Technical Report No. 296, North Carolina State University, Raleigh, North Carolina, 1994, submitted to SIAM Journal on Optimization.
Mishra, S., and Fang, S.-C., “A Maximum Entropy Optimization Approach to Tandem Queues with Generalized Blocking,” Performance Evaluation, Vol. 702, 1997, pp. 1–25.
Oko, S.O., “Surrogate Methods for Linear Inequalities,” Journal of Optimization Theory and Applications, Vol. 72, 1992, pp. 247–268.
Polak, E., Mayne, D.Q., and Higgins, J.E., “Superlinearly Convergent Algorithm for Min-Max Problems,” Journal of Optimization Theory and Applications, Vol. 69, 1991, pp. 407–439.
Polyak, R.A., “Smooth Optimization Methods for Minimax Problems,” SIAM Journal of Control and Optimization, Vol. 26, 1988, pp. 1274–1286.
Rajasekera, J.R., and Fang, S.-C., “Deriving an Unconstrained Convex Program for Linear Programming,” Journal of Optimization Theory and Applications, Vol. 75, 1992, pp. 603–612.
Royden, H.L., Real Analysis, 2nd Edition, The Macmillan Company, New York, 1972.
Rudin, W., Principles of Mathematical Analysis, McGraw Hill, New York, 1976.
Tsao, H.-S.J., Fang, S.-C., and Lee, D.N., “On the Optimal Entropy Analysis,” European Journal of Operational Research, Vol. 59, 1992, pp. 324–329.
Tsao, H.-S.J., Fang, S.-C., and Lee, D.N., “A Bayesian Interpretation of the Linearly-Constrained Cross-Entropy Minimization Problem,” Engineering Optimization, Vol. 22, 1993, pp. 65–75.
Vardi, A., “New Minmax Algorithm,” Journal of Optimization Theory and Applications, Vol. 75, 1992, pp. 613–633.
Williams, P.M., “Bayesian Conditionalisation and the Principle of Minimum Information,” The British Journal for the Philosophy of Science, Vol. 31, 1980, pp. 131–144.
Wu, J.-S., and Chan, W.C., “Maximum Entropy Analysis of Multiple-Server Queueing Systems,” Journal of Operational Research Society, Vol. 40, 1989, pp. 815–826.
Yang, K., and Murty, K.G., “New Iterative Methods for Linear Inequalities,” Journal of Optimization Theory and Applications, Vol. 72, 1992, pp. 163–185.
Zang, I., “A Smoothing Out Technique for Min-Max Optimization,” Mathematical Programming, Vol. 19, 1980, pp. 61–77.
Zellner, A., “Optimal Information Processing and Bayes’ Theorem,” The American Statistician, Vol. 42, 1988, pp. 278–284.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer Science+Business Media New York
About this chapter
Cite this chapter
Fang, SC., Rajasekera, J.R., Tsao, HS.J. (1997). Extensions and Related Results. In: Entropy Optimization and Mathematical Programming. International Series in Operations Research & Management Science, vol 8. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-6131-6_7
Download citation
DOI: https://doi.org/10.1007/978-1-4615-6131-6_7
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-7810-5
Online ISBN: 978-1-4615-6131-6
eBook Packages: Springer Book Archive