Log-Polynomial Optimization

TL;DR

This work addresses global optimization of log-polynomial objectives $\max_{x\in K} \sum_{i=1}^m a_i\log p_i(x)$ under polynomial constraints by extending the Moment-SOS framework through a hierarchy of truncated $K$-moment relaxations. It establishes convergence and tightness criteria, provides flat-truncation based strategies to extract global optimizers, and introduces Lagrange multiplier expressions to construct tighter relaxations for box, simplex, and ball constraints. The authors demonstrate theoretical results for (i) tightness conditions, (ii) SOS-concave cases guaranteeing finite convergence, and (iii) practical tight relaxations via LMEs, with numerical experiments on maximum likelihood-like problems and entropy-based losses. The approach offers global optimization guarantees for log-polynomial estimations in statistics and machine learning, and points to scalable extensions leveraging sparsity and nonpolynomial dynamics.

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.

Theorems & Definitions (25)

• Example 3.1
• Proposition 3.2
• proof
• Corollary 3.3
• Theorem 3.4
• proof
• Corollary 3.5
• Theorem 4.1
• proof
• Corollary 4.2