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An Efficient Framework for Global Non-Convex Polynomial Optimization with Algebraic Constraints

Mitchell Tong Harris, Pierre-David Letourneau, Dalton Jones, M. Harper Langston

TL;DR

This work tackles constrained global non-convex polynomial optimization over the hypercube by introducing a generalized nonlinear reformulation based on moment representations and products of measures. By enforcing algebraic constraints through moment- and sum-of-squares–style conditions and leveraging a Burer–Monteiro factorization to handle semidefinite constraints, the approach yields an equivalent reformulation with essentially no spurious local minima, enabling descent-based methods to reach global optima. Theoretical guarantees establish equivalence of the reformulation with the original problem and a construction that recovers optimal solutions from the reformulated moments. Numerical experiments on high-dimensional, previously intractable problems (elliptical annulus and discrete-feasible regions) demonstrate polynomial scaling in dimension and degree for computing both the optimal value and the optimizer location, with the reformulation outperforming the original formulation by avoiding traps in nonconvex landscapes. The framework offers a practical, scalable pathway for solving polynomial optimization problems with algebraic constraints in applications requiring global guarantees.

Abstract

We present an efficient framework for solving algebraically-constrained global non-convex polynomial optimization problems over subsets of the hypercube. We prove the existence of an equivalent nonlinear reformulation of such problems that possesses essentially no spurious local minima. Through numerical experiments on previously intractable global constrained polynomial optimization problems in high dimension, we show that polynomial scaling in dimension and degree is achievable when computing the optimal value and location.

An Efficient Framework for Global Non-Convex Polynomial Optimization with Algebraic Constraints

TL;DR

This work tackles constrained global non-convex polynomial optimization over the hypercube by introducing a generalized nonlinear reformulation based on moment representations and products of measures. By enforcing algebraic constraints through moment- and sum-of-squares–style conditions and leveraging a Burer–Monteiro factorization to handle semidefinite constraints, the approach yields an equivalent reformulation with essentially no spurious local minima, enabling descent-based methods to reach global optima. Theoretical guarantees establish equivalence of the reformulation with the original problem and a construction that recovers optimal solutions from the reformulated moments. Numerical experiments on high-dimensional, previously intractable problems (elliptical annulus and discrete-feasible regions) demonstrate polynomial scaling in dimension and degree for computing both the optimal value and the optimizer location, with the reformulation outperforming the original formulation by avoiding traps in nonconvex landscapes. The framework offers a practical, scalable pathway for solving polynomial optimization problems with algebraic constraints in applications requiring global guarantees.

Abstract

We present an efficient framework for solving algebraically-constrained global non-convex polynomial optimization problems over subsets of the hypercube. We prove the existence of an equivalent nonlinear reformulation of such problems that possesses essentially no spurious local minima. Through numerical experiments on previously intractable global constrained polynomial optimization problems in high dimension, we show that polynomial scaling in dimension and degree is achievable when computing the optimal value and location.
Paper Structure (25 sections, 7 theorems, 60 equations, 6 figures, 3 tables)

This paper contains 25 sections, 7 theorems, 60 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Theory
  4. Preliminaries
  5. Polynomials
  6. Moments of measures
  7. Moment matrices
  8. Reformulation on the hypercube
  9. Generalized reformulation for polynomial constraints
  10. Semi-Algebraic Constraints
  11. Algorithm and Implementation
  12. Handling Semidefinite Constraints
  13. Numerical Implementation Details
  14. Numerical experiments
  15. Elliptical Annulus
  16. ...and 10 more sections

Key Result

Proposition 0

Let $D,d \in \mathbb{N}$ and $\mu := ( \mu_1, \mu_2, ..., \mu_D ) \in \mathbb{R}^{(2d+1) \times D}$ be such that for each $i = 1, ..., D$, $\mu_{i,0} = 1$, and and in addition for every $j=1, ..., J$, where $\gamma^{(j)}$ is the vector of polynomial coefficients or $\left ( g^{(j)}(x) \right )^2$. Then there exists a regular Borel product measure, supported over the algebraic set such that $\m

Figures (6)

  • Figure 1: Representation of Problem \ref{['eqn:ellipticalannulus']} in two dimensions (2D). The feasible region consists of non-convex spherical shells. The objective is concave in the first dimension, constant in remaining dimensions, and possesses four (4) local minima over the feasible set.
  • Figure 2: Average relative errors in spherical shells problem. The error is on average approximately $10^{-4}$ and lies below the solver threshold ($10^{-2}$) regardless of the dimension $D$.
  • Figure 3: Average wall times of solving reformulated elliptical annulus problem using our novel formulation (solid line). The dotted line indicates a scaling of $O(D^{5})$.
  • Figure 4: Two dimensional version of discrete non-convex optimization problem.
  • Figure 5: Average relative objective errors in disjoint patches problem. The error is on average approximately $10^{-2}$ and lies below the solver threshold ($10^{-1}$) regardless of the dimension $D$.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Proposition 0
  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5: Proposition 1, Letourneau2023unconstrained
  • Proposition 5
  • proof