Leveraging Christoffel-Darboux Kernels to Strengthen Moment-SOS Relaxations

TL;DR

This work addresses the scalability bottleneck of the Moment-SOS hierarchy in polynomial optimization by introducing Christoffel-Darboux kernel-based constraints that strengthen a fixed-order relaxation. It formulates two practical heuristics, H1 (iterative feasibility-shrinking) and H2 (local-solution guided), to tighten the order-$d$ bound without solving higher-order SDPs, and extends them to correlative sparsity via H1CS and H2CS. The methods exploit CDK sublevel sets and marginal CD polynomials to confine the feasible region and guide improvement, achieving substantial gap reductions and often enabling minimizer extraction with a fraction of the cost of higher-order relaxations. These techniques offer a versatile, scalable avenue to accelerate POP solvers and could be integrated into Branch & Bound or extended to more general discrete constraints, broadening the practical impact of SDP-based optimization.

Abstract

The classical Moment-Sum Of Squares hierarchy allows to approximate a global minimum of a polynomial optimization problem through semidefinite relaxations of increasing size. However, for many optimization instances, solving higher order relaxations becomes impractical or even impossible due to the substantial computational demands they impose. To address this, existing methods often exploit intrinsic problem properties, such as symmetries or sparsity. Here, we present a complementary approach, which enhances the accuracy of computationally more efficient low-order relaxations by leveraging Christoffel-Darboux kernels. Such strengthened relaxations often yield significantly improved bounds or even facilitate minimizer extraction. We illustrate the efficiency of our approach on several classes of important quadratically constrained quadratic Programs.

Figures (5)

• Figure 1: Sublevel sets of the Christoffel polynomials associated with the uniform measure over the unit square, i.e., $d\mu(\bm{x})=\mathbbm{1}_{[0,1]^2}(\bm{x})d\bm{x}$. The figure depicts $S_1(\mu,\gamma)$ (left) and $S_2(\mu,\gamma)$ (right) for $\gamma \in \{5, 7, 26, 40, 100\}$.
• Figure 2: Depicting sublevel sets $\widetilde{S}_1(\bm{y}^{*},\gamma)$ associated to the POP from Example \ref{['ex: Leitmotif']}, where $\gamma\in\left\{1.01, 1.15, 1.50, 2.0, 2.85, 3.0\right\}$. The red point is the true minimizer of $f$, and the blue point corresponds to the pseudo-moments of order one extracted from the optimal solution $\bm{y}^*$ of the first-order moment relaxation.
• Figure 3: Different iterations $k\in\left\{0,5,15,25\right\}$ of Algorithm \ref{['alg:iteration-based-heuristic']}, with $\varepsilon=0.95$.
• Figure 4: Algorithm \ref{['alg:loc-sol-based-heuristic']} applied to Example \ref{['ex: Leitmotif']} for different values of $\tau$. The blue point is $\hat{\bm{x}}=(1.6562,2.0833)$, and the red point corresponds to $\overline{\bm{x}}=\bm{x}^{*}=(2,2)$.
• Figure 5: Trajectories $(\widetilde{f}_{1,k})_{0 \leq k \leq 15}$ from Algorithm \ref{['alg:iteration-based-heuristic']} for different values of $\varepsilon$, in dimension $n=20$, with $(\mathbf{Q},\bm{q})$ dense and randomly generated, using seeds 16 (left) and 38 (right).

Theorems & Definitions (13)

• Proposition 2.1
• proof
• Theorem 2.2: lasserre2001global
• Theorem 2.3: Theorem 1.6, curto2000truncated
• Example 2.4
• Remark 2.5
• Proposition 3.1
• proof
• Remark 3.1
• Proposition 3.2