# Signomial and Polynomial Optimization via Relative Entropy and Partial Dualization

## TL;DR

This paper introduces a generalized SAGE relaxation method for bounding constrained signomial and polynomial optimization problems, leveraging relative entropy and convex duality for improved solution insights.

## Contribution

It extends SAGE relaxations to a broader class of problems using relative entropy, offering better transparency and a new perspective-based solution recovery method.

## Key findings

- Effective bounds on complex optimization problems
- Preserves sparsity in solutions
- Demonstrated utility with examples and software

## Abstract

We describe a generalization of the Sums-of-AM/GM Exponential (SAGE) relaxation methodology for obtaining bounds on constrained signomial and polynomial optimization problems. Our approach leverages the fact that relative entropy based SAGE certificates conveniently and transparently blend with convex duality, in a manner that Sums-of-Squares certificates do not. This more general approach not only retains key properties of ordinary SAGE relaxations (e.g. sparsity preservation), but also inspires a novel perspective-based method of solution recovery. We illustrate the utility of our methodology with a range of examples from the global optimization literature, along with a publicly available software package.