Probabilistic Algorithm for Polynomial Optimization over a Real Algebraic Set

TL;DR

This paper presents a probabilistic algorithm for exact polynomial optimization over real algebraic sets, capable of computing the global infimum and solutions under regularity assumptions, with practical efficiency demonstrated through implementation.

Contribution

It introduces a novel probabilistic approach using polar varieties for polynomial optimization, achieving complexity comparable to deterministic methods while enabling practical solutions.

Findings

Algorithm computes exact infimum and solutions for polynomial optimization.

Implementation can handle problems previously unsolvable by exact algorithms.

First probabilistic method with practical efficiency and controlled complexity for this problem.

Abstract

Let $f, f_1, \ldots, f_\nV$ be polynomials with rational coefficients in the indeterminates $\bfX=X_1, \ldots, X_n$ of maximum degree $D$ and $V$ be the set of common complex solutions of $\F=(f_1,\ldots, f_\nV)$. We give an algorithm which, up to some regularity assumptions on $\F$, computes an exact representation of the global infimum $f^\star=\inf_{x\in V\cap\R^n} f\Par{x}$, i.e. a univariate polynomial vanishing at $f^\star$ and an isolating interval for $f^\star$. Furthermore, this algorithm decides whether $f^\star$ is reached and if so, it returns $x^\star\in V\cap\R^n$ such that $f\Par{x^\star}=f^\star$. This algorithm is probabilistic. It makes use of the notion of polar varieties. Its complexity is essentially cubic in $\Par{\nV D}^n$ and linear in the complexity of evaluating the input. This fits within the best known deterministic complexity class $D^{O(n)}$. We report on some practical experiments of a first implementation that is available as a Maple package. It appears that it can tackle global optimization problems that were unreachable by previous exact algorithms and can manage instances that are hard to solve with purely numeric techniques. As far as we know, even under the extra genericity assumptions on the input, it is the first probabilistic algorithm that combines practical efficiency with good control of complexity for this problem.