We consider a hierarchy of upper approximations for the minimization of a polynomial f over a compact set $$K \subseteq \mathbb {R}^n$$ proposed recently by Lasserre (arXiv:1907.097784, 2019). This hierarchy relies on using the push-forward measure of the Lebesgue measure on K by the polynomial f and involves univariate sums of squares of polynomials with growing degrees 2r. Hence it is weaker, but cheaper to compute, than an earlier hierarchy by Lasserre (SIAM Journal on Optimization 21(3), 864–885, 2011), which uses multivariate sums of squares. We show that this new hierarchy converges to the global minimum of f at a rate in $$O(\log ^2 r / r^2)$$ whenever K satisfies a mild geometric condition, which holds, eg., for convex bodies and for compact semialgebraic sets with dense interior. As an application this rate of convergence also applies to the stronger hierarchy based on multivariate sums of squares, which improves and extends earlier convergence results to a wider class of compact sets. Furthermore, we show that our analysis is near-optimal by proving a lower bound on the convergence rate in $$\varOmega (1/r^2)$$ for a class of polynomials on $$K=[-1,1]$$ , obtained by exploiting a connection to orthogonal polynomials.

中文翻译：

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用于多元多项式优化的 Lasserre 基于单变量度量的边界的近最优分析

我们考虑最近由 Lasserre (arXiv:1907.097784, 2019) 提出的紧致集合 $$K \subseteq \mathbb {R}^n$$ 上多项式 f 的最小化的上近似层次。该层次结构依赖于通过多项式 f 对 K 使用 Lebesgue 测度的前推测度，并涉及增长次数为 2r 的多项式的单变量平方和。因此，与 Lasserre 的早期层次结构（SIAM Journal on Optimization 21(3), 864–885, 2011）相比，它更弱，但计算成本更低，后者使用多元平方和。我们表明，每当 K 满足温和的几何条件时，这个新的层次就会以 $$O(\log ^2 r / r^2)$$ 的速率收敛到 f 的全局最小值，例如，对于凸体和具有稠密内部的紧半代数集。作为一种应用，这种收敛速度也适用于基于多元平方和的更强层次结构，这改进了早期收敛结果并将其扩展到更广泛的紧凑集类。此外，我们通过证明 $$\varOmega (1/r^2)$$ 中的一类多项式在 $$K=[-1,1] 上的收敛速度的下界来表明我们的分析接近最优$$ ，通过利用与正交多项式的连接获得。