Multi-Degrees in Polynomial Optimization

TL;DR

This paper investigates polynomial optimization problems with multi-degree structures, analyzing the number of critical points to understand their algebraic complexity and exploring numerical methods for solving and certifying solutions.

Contribution

It introduces a multi-degree framework for polynomial optimization and determines the generic number of critical points, advancing understanding of algebraic complexity and solution certification.

Findings

Determined the generic number of complex critical points for fixed multi-degrees.

Provided methods for computation and certification from numerical nonlinear algebra.

Enhanced understanding of algebraic complexity in structured polynomial optimization.

Abstract

We study structured optimization problems with polynomial objective function and polynomial equality constraints. The structure comes from a multi-grading on the polynomial ring in several variables. For fixed multi-degrees we determine the generic number of complex critical points. This serves as a measure for the algebraic complexity of the optimization problem. We also discuss computation and certification methods coming from numerical nonlinear algebra.