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An Efficient Framework for Global Non-Convex Polynomial Optimization over the Hypercube

Pierre-David Letourneau, Dalton Jones, Matthew Morse, M. Harper Langston

TL;DR

The paper tackles the challenge of global optimization for non-convex polynomials on the hypercube $[-1,1]^D$ by introducing a reformulation that casts the problem as a nonlinear objective over a convex cone of semidefinite moment matrices with fixed, polynomially-bounded size. It proves equivalence of the reformulated problem to the original and, for $L\ge 2$, guarantees the absence of spurious local minima, enabling efficient descent-based optimization via a Burer–Monteiro–augmented-Lagrangian algorithm. Theoretical results are supported by a measure-extension argument and by a constructive descent framework that connects local minima of the reformulation to global minima of the original problem. Numerical experiments on two challenging polynomial classes demonstrate correct recovery of global minima with near-zero relative error up to large dimensions, and reveal favorable polynomial scaling (quadratic in Example 1, quartic in Example 2). The work offers a scalable path to global polynomial optimization in high dimensions and lays groundwork for extensions to semi-algebraic constraints and parallelized implementations.

Abstract

We present a novel efficient theoretical and numerical framework for solving global non-convex polynomial optimization problems. We analytically demonstrate that such problems can be efficiently reformulated using a non-linear objective over a convex set; further, these reformulated problems possess no spurious local minima (i.e., every local minimum is a global minimum). We introduce an algorithm for solving these resulting problems using the augmented Lagrangian and the method of Burer and Monteiro. We show through numerical experiments that polynomial scaling in dimension and degree is achievable for computing the optimal value and location of previously intractable global polynomial optimization problems in high dimension.

An Efficient Framework for Global Non-Convex Polynomial Optimization over the Hypercube

TL;DR

The paper tackles the challenge of global optimization for non-convex polynomials on the hypercube by introducing a reformulation that casts the problem as a nonlinear objective over a convex cone of semidefinite moment matrices with fixed, polynomially-bounded size. It proves equivalence of the reformulated problem to the original and, for , guarantees the absence of spurious local minima, enabling efficient descent-based optimization via a Burer–Monteiro–augmented-Lagrangian algorithm. Theoretical results are supported by a measure-extension argument and by a constructive descent framework that connects local minima of the reformulation to global minima of the original problem. Numerical experiments on two challenging polynomial classes demonstrate correct recovery of global minima with near-zero relative error up to large dimensions, and reveal favorable polynomial scaling (quadratic in Example 1, quartic in Example 2). The work offers a scalable path to global polynomial optimization in high dimensions and lays groundwork for extensions to semi-algebraic constraints and parallelized implementations.

Abstract

We present a novel efficient theoretical and numerical framework for solving global non-convex polynomial optimization problems. We analytically demonstrate that such problems can be efficiently reformulated using a non-linear objective over a convex set; further, these reformulated problems possess no spurious local minima (i.e., every local minimum is a global minimum). We introduce an algorithm for solving these resulting problems using the augmented Lagrangian and the method of Burer and Monteiro. We show through numerical experiments that polynomial scaling in dimension and degree is achievable for computing the optimal value and location of previously intractable global polynomial optimization problems in high dimension.
Paper Structure (19 sections, 16 theorems, 128 equations, 6 figures, 3 tables, 2 algorithms)

This paper contains 19 sections, 16 theorems, 128 equations, 6 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Notation
  4. Efficient Reformulation for Global Non-Convex Polynomial Optimization Problems
  5. Theoretical Results
  6. Numerical Results
  7. Numerical Example 1
  8. Numerical Example 2
  9. Conclusion
  10. Proofs
  11. Algorithm
  12. Optimal Location
  13. Further Implementation details
  14. Constraint simplification
  15. Numerical stability
  16. ...and 4 more sections

Key Result

Proposition 1

\newlabelprop:measureextension0 Let $D,d \in \mathbb{N}$ and $( \mu_1, \mu_2, ..., \mu_D ) \in \mathbb{R}^{(2d+1) \times D}$ be such that for each $i = 1, ..., D$, $\mu_{i,0} = 1$, and Then, there exists a regular Borel product measure, supported over $[-1,1]^D$ such that $\mu \left ( [-1,1]^D \right ) = 1$, and for all multi-index ${\bf n} \in \mathbb{N}^D$ such that $0 \leq {\bf n}_i \leq

Figures (6)

  • Figure 1: Two-dimensional (2-D) rendering of the non-convex function $f_2(x)$ in Example \ref{['sec:numerical_example_1']}. For the exponential number of local minima, global optimization of this function would be intractable using traditional techniques.
  • Figure 2: Relative error vs problem dimension ($D$) for dimension $1$ to $250$ for Example \ref{['sec:numerical_example_1']}. Top: relative error associated with computed optimal value, Bottom: relative error associated with computed optimal location. The error lies below the expected upper bound (black dotted line) of $10^{-2}$ in every case.
  • Figure 3: Computation (wall) time (s) vs problem dimension ($D$) for Example \ref{['sec:numerical_example_1']}. Our numerical implementation exhibits polynomial (quadratic; (black line)) scaling on problems previously considered intractable.
  • Figure 4: Two-dimensional (2-D) rendering of the non-convex function $g_2(x)$ from Example \ref{['sec:numerical_example_2']}. For an exponential number of local minima, as well as the asymmetry in this function, global optimization is intractable using traditional techniques.
  • Figure 5: Relative error vs problem dimension ($D$) for dimension $1$ to $45$ for Example \ref{['sec:numerical_example_2']}. Top: optimal value; Bottom: optimal location. The error lies below the expected upper bound (dotted black line) of $10^{-2}$ in every case.
  • ...and 1 more figures

Theorems & Definitions (24)

  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Theorem 1: Powers, Reznick, from Fekete powers2000polynomials
  • Theorem 2: Riesz extension theorem, rudin1986real
  • Theorem 3: Riesz-Markov,reed1980functional
  • Theorem 4: B.L.T., reed1980functional
  • Proposition 2: Stone-Weierstrass (corollary), reed1980functional
  • Proposition 2
  • Proof 1
  • ...and 14 more