Connecting Max-entropy With Computational Geometry, LP And SDP
Jean B Lasserre
TL;DR
The paper establishes a rigorous bridge between maximum entropy (KL-Maxent) problems with generalized moment constraints and canonical linear and semidefinite programs. By leveraging Federer's coarea formula, it shows that, when the number of moment constraints $m$ is at most the ambient dimension $d$, the KL-Maxent objective $\Theta(y)$ is the Cramér transform of a $(d-m)$-dimensional density $v$—the density of the pushforward $\mathbf{h}_\#P$. It then proves that, with appropriate reference measures, the perspective $\varepsilon\Theta(y/\varepsilon)$ coincides (up to constants) with the log-barrier formulations of the dual LP (and SDP) as $\varepsilon\to 0$, thereby recovering the LP (or SDP) optimum values in the limit. The results unify optimization and integration perspectives, revealing a concrete computational pathway from entropy-based methods to standard convex optimization tools, with explicit representations for the partition function via $\mathcal{L}_v$. These connections also provide constructive interpretations of $Z(\lambda)$ and offer potential computational avenues through known geometry-based formulas (e.g., Brion–Vergnes) for the LP case.
Abstract
We consider the well-known max-(relative) entropy problem $Θ$(y) = infQ$\ll$P DKL(Q P ) with Kullback-Leibler divergence on a domain $Ω$ $\subset$ R d , and with ''moment'' constraints h dQ = y, y $\in$ R m . We show that when m $\le$ d, $Θ$ 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 $ε$ $Θ$(y/$ε$) is the optimal value of the log-barrier formulation (with parameter $ε$) of the dual LP (and so it converges to the LP optimal value as $ε$ $\rightarrow$ 0). An analogous result also holds for the canonical SDP: min X 0 { C, X\,: A(X) = y }.