Abstract
Quadratically constrained minimum cross-entropy problem has recently been studied by Zhang and Brockett through an elaborately constructed dual. In this paper, we take a geometric programming approach to analyze this problem. Unlike Zhang and Brockett, we separate the probability constraint from general quadratic constraints and use two simple geometric inequalities to derive its dual problem. Furthermore, by using the dual perturbation method, we directly prove the “strong duality theorem” and derive a “dual-to-primal” conversion formula. As a by-product, the perturbation proof gives us insights to develop a computation procedure that avoids dual non-differentiability and allows us to use a general purpose optimizer to find anε-optimal solution for the quadratically constrained minimum cross-entropy analysis.
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References
A. Charnes and W.W. Cooper, “An extremal principle for accounting balance of a resource-value transfer economy: existence, uniqueness and computation,”Rendiconte Accademia Nazionale dei Lincei, Rome (8) 4 (1974) 556–578.
A. Charnes, W.W. Cooper and L. Seiford, “Extremal principles and optimization qualities for Khinchin-Kullback-Leibler estimation,”Statistics (Zentralinstitut für Mathematik und Methanik) 9 (1978) 21–29.
R.J. Duffin, E.L. Peterson and C. Zener,Geometric Programming—Theory and Applications (John Wiley, New York, 1967).
S.C. Fang and J.R. Rajasekera, “Controlled perturbations for quadratically constrained quadratic programs,”Mathematical Programming 36 (1986) 276–289.
S.C. Fang and J.R. Rajasekera, Controlled dual perturbations forl p -programming,”Zeitschrift für Operations Research 30 (1986) A29-A42.
S. Guiasu,Information Theory with Applications (McGraw-Hill, New York, 1977).
S. Kullback,Information Theory and Statistics (John Wiley, New York, 1959).
B.A. Murtagh and M.A. Saunders, “MINOS users guide,” Technical Report 77–9, Systems Optimization Laboratory, Stanford University (1977).
R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, 1970).
J.E. Shore, “Minimum cross-entropy spectral analysis,”IEEE Transactions on Acoustics, Speech, and Signal Processing 29(2) (1981) 230–237.
J.E. Shore and R.W. Johnson, “Axiomatic derivation of principle of maximum entropy and principle of minimum cross-entropy,”IEEE Transactions on Information Theory 26(1) (1980) 26–37.
J. Zhang and P.L. Brockett, “Quadratically constrained information theoretic analysis,”SIAM Journal on Applied Mathematics 47 (1987) 871–885.
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Fang, S.C., Rajasekera, J.R. Quadratically constrained minimum cross-entropy analysis. Mathematical Programming 44, 85–96 (1989). https://doi.org/10.1007/BF01587079
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DOI: https://doi.org/10.1007/BF01587079