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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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