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Convergence rates of S.O.S hierarchies for polynomial semidefinite programs

Hoang Anh Tran, Kim-Chuan Toh

TL;DR

This work develops a streamlined sum-of-squares (SOS) hierarchy for polynomial optimization problems constrained by a polynomial matrix inequality $G(x)\succeq0$. By introducing a penalty framework that replaces the matrix constraint with a scalar, nonnegative trace term, the authors reduce the SDP size relative to Kronecker-based formulations and obtain explicit convergence rates for both discrete and continuous feasible sets. They establish a Putinar-type Positivstellensatz in the matrix setting, with degree bounds derived via Jackson approximation and a matrix Łojasiewicz inequality, yielding polynomial-in-$m$ rate bounds and, in special cases, exponential-type decay in the hierarchy index $r$. The approach delivers practical SDP relaxations for binary and ball-contained problems, avoiding excessive growth in matrix sizes while providing rigorous guarantees on convergence and degree. Overall, the paper extends scalar SOS theory to matrix-defined feasible sets with quantifiable rates and scalable SDP formulations, offering new tools for matrix-POP benchmarks and control/optimization applications.

Abstract

We introduce an S.o.S hierarchy of lower bounds for a polynomial optimization problem whose constraint is expressed as a matrix polynomial semidefinite inequality. Our approach involves utilizing a penalty function framework to directly address the matrix-based constraint, making it applicable to both discrete and continuous polynomial optimization problems. We investigate the convergence rates of these bounds in both types of problems. The proposed method yields a variant of Putinar's theorem, tailored for positive polynomials within a compact semidefinite set $\mathcal{X}$ defined by a matrix polynomial semidefinite constraint. More specifically, we derive novel insights into the convergence rates and bounds on the degree of the S.o.S polynomials required to certify positivity on $\mathcal{X}$, based on Jackson's theorem and a variant of the Łojasiewicz inequality.

Convergence rates of S.O.S hierarchies for polynomial semidefinite programs

TL;DR

This work develops a streamlined sum-of-squares (SOS) hierarchy for polynomial optimization problems constrained by a polynomial matrix inequality . By introducing a penalty framework that replaces the matrix constraint with a scalar, nonnegative trace term, the authors reduce the SDP size relative to Kronecker-based formulations and obtain explicit convergence rates for both discrete and continuous feasible sets. They establish a Putinar-type Positivstellensatz in the matrix setting, with degree bounds derived via Jackson approximation and a matrix Łojasiewicz inequality, yielding polynomial-in- rate bounds and, in special cases, exponential-type decay in the hierarchy index . The approach delivers practical SDP relaxations for binary and ball-contained problems, avoiding excessive growth in matrix sizes while providing rigorous guarantees on convergence and degree. Overall, the paper extends scalar SOS theory to matrix-defined feasible sets with quantifiable rates and scalable SDP formulations, offering new tools for matrix-POP benchmarks and control/optimization applications.

Abstract

We introduce an S.o.S hierarchy of lower bounds for a polynomial optimization problem whose constraint is expressed as a matrix polynomial semidefinite inequality. Our approach involves utilizing a penalty function framework to directly address the matrix-based constraint, making it applicable to both discrete and continuous polynomial optimization problems. We investigate the convergence rates of these bounds in both types of problems. The proposed method yields a variant of Putinar's theorem, tailored for positive polynomials within a compact semidefinite set defined by a matrix polynomial semidefinite constraint. More specifically, we derive novel insights into the convergence rates and bounds on the degree of the S.o.S polynomials required to certify positivity on , based on Jackson's theorem and a variant of the Łojasiewicz inequality.
Paper Structure (19 sections, 21 theorems, 158 equations, 2 figures)

This paper contains 19 sections, 21 theorems, 158 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Hierarchy of lower bounds in the matrix case
  3. Related works
  4. Contribution
  5. Preliminary
  6. Hierarchy of lower bounds in the scalar case
  7. Moment matrix and localizing matrix
  8. Convergence rates of moment-SOS hierarchies
  9. Methodology
  10. Binary polynomial optimization problems with polynomial matrix semidefinite constraints
  11. Matrix moment-SOS hierarchy
  12. Simple representations of $\mathcal{X}$
  13. Approximation of a nonnegative piecewise affine function by nonnegative polynomials
  14. A Hierarchy for Binary Polynomial Optimization Problems
  15. Continuous Polynomial Optimization Problems with semidefinite constraints
  16. ...and 4 more sections

Key Result

Theorem 1.1

hol2004sum \newlabelGPutinar0 Suppose $\mathcal{X}$ satisfies the Archimedean condition, that is, there exist an SOS polynomial $\sigma$, an SOS polynomial matrix $R(x)$, and a scalar $N$ such that Then every positive polynomial $f$ on $\mathcal{X}$ belongs to the quadratic module $Q(\mathcal{X})$.

Figures (2)

  • Figure 1: The left panel shows the polynomial approximation of $a(t)$, and the right panel shows the polynomial approximation of $a(t)$ from below after a vertical translation.
  • Figure 2: Comparison of $c_0, c_1,c_2,c_3$ and $c_4$ over the interval $[0,1]$ in terms of their smoothness at the end points of the interval.

Theorems & Definitions (45)

  • Theorem 1.1
  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Proposition 3.1
  • Proof 1
  • ...and 35 more