Quadratically constrained minimum cross-entropy analysis

Quadratically constrained minimum cross-entropy problem has recently been studied by Zhang and Brockett through an elaborately constructed dual. In this paper, the authors take a geometric programming approach to analyze this problem. Unlike Zhang and Brockett, they 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, the authors directly prove the ‘strong duality theorem’ and derive a ‘dual-to-primal’ conversion formula. As a by-product, the perturbation proof gives insights to develop a computation procedure that avoids dual non-differentiability and allows them to use a general purpose optimizer to find an •-optimal solution for the quadratically constrained minimum cross-entropy analysis.