Connecting Max-entropy With Computational Geometry, LP And SDP

Abstract

We consider the well-known max-(relative) entropy problem $\hat{I}˜$(y) = infQ$\ll$P DKL(Q P ) with Kullback-Leibler divergence on a domain $\hat{I}\text{c◯}$ $\subset$ R d , and with ''moment'' constraints h dQ = y, y $\in$ R m . We show that when m $\le$ d, $\hat{I}˜$ is the Cram{é}r transform of a function v that solves a simply related computational geometry problem. Also, and remarkably, to the canonical LP: min x$\ge$0 {c T x\,: A x = y}, with A $\in$ R mxd , one may associate a max-entropy problem with a suitably chosen reference measure P on R d + and linear mapping h(x) = Ax, such that its associated perspective function $\hat{I}\mu$ $\hat{I}˜$(y/$\hat{I}\mu$) is the optimal value of the log-barrier formulation (with parameter $\hat{I}\mu$) of the dual LP (and so it converges to the LP optimal value as $\hat{I}\mu$ $\to$ 0). An analogous result also holds for the canonical SDP: min X 0 { C, X\,: A(X) = y }.