Log-Polynomial Optimization

TL;DR

This paper introduces a hierarchy of moment relaxations for solving log-polynomial optimization problems, motivated by statistical estimation, and demonstrates their effectiveness through applications and numerical experiments.

Contribution

It proposes a novel hierarchy of relaxations based on truncated K-moment problems for log-polynomial optimization, with conditions for tightness and methods for extracting global solutions.

Findings

Hierarchy of relaxations effectively solves log-polynomial problems.

Conditions for relaxation tightness are established.

Numerical experiments show high efficiency and applicability.

Abstract

We study an optimization problem in which the objective is given as a sum of logarithmic-polynomial functions. This formulation is motivated by statistical estimation principles such as maximum likelihood estimation, and by loss functions including cross-entropy and Kullback-Leibler divergence. We propose a hierarchy of moment relaxations based on the truncated $K$-moment problems to solve log-polynomial optimization. We provide sufficient conditions for the hierarchy to be tight and introduce a numerical method to extract the global optimizers when the tightness is achieved. In addition, we modify relaxations with optimality conditions to better fit log-polynomial optimization with convenient Lagrange multipliers expressions. Various applications and numerical experiments are presented to show the efficiency of our method.