# An effective version of Schmüdgen’s Positivstellensatz for the hypercube

@article{Laurent2022AnEV,
title={An effective version of Schm{\"u}dgen’s Positivstellensatz for the hypercube},
author={Monique Laurent and Lucas Slot},
journal={Optimization Letters},
year={2022}
}
• Published 20 September 2021
• Art
• Optimization Letters
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## References

SHOWING 1-10 OF 26 REFERENCES

### Error bounds for polynomial optimization over the hypercube using putinar type representations

This work gives new bounds for Putinar type representations by relating the quadratic module and the preordering associated with the polynomials to the Positivstellensatz of Putinar.

### An Approximation Bound Analysis for Lasserre’s Relaxation in Multivariate Polynomial Optimization

It is proved that under a suitable condition on g1,⋯,gm, it is proven that (f_{\max}- f_{\mathrm{sos}}) \leqslant Q(f-f fmin) is a constant depending only on f but not on f, with Q a constantdepending only on g 1,Ⓜ⓽,gm butNot on f.

### Complexity Estimates for the Schmu"dgen Positivstellensatz

LetKbe a closed basic set inRngiven by the polynomial inequalities ?1? 0, ... , ?m? 0 and let ? be the semiring generated by the ?kand the squares inRx1, ... ,xn]. Schmudgen has shown that ifKis

### Sum-of-squares hierarchies for binary polynomial optimization

• Mathematics, Computer Science
IPCO
• 2021
The proof combines classical Fourier analysis on B n with the polynomial kernel technique and existing results on the extremal roots of Krawtchouk polynomials to establish the worst-case error in the regime.

### Improved Convergence Rates for Lasserre-Type Hierarchies of Upper Bounds for Box-Constrained Polynomial Optimization

• Mathematics, Computer Science
SIAM J. Optim.
• 2017
We consider the problem of minimizing a given $n$-variate polynomial $f$ over the hypercube $[-1,1]^n$. An idea introduced by Lasserre, is to find a probability distribution on $[-1,1]^n$ with

### Near-optimal analysis of univariate moment bounds for polynomial optimization

• Mathematics
• 2020
We consider a recent hierarchy of upper approximations proposed by Lasserre (arXiv:1907.097784, 2019) for the minimization of a polynomial $f$ over a compact set $K \subseteq \mathbb{R}^n$. This

### Near-optimal analysis of Lasserre's univariate measure-based bounds for multivariate polynomial optimization

• Mathematics
Math. Program.
• 2021
It is shown that this new hierarchy based on multivariate sums of squares, which improves and extends earlier convergence results to a wider class of compact sets, 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 polynmials.

### On the complexity of Putinar's Positivstellensatz

• Mathematics
J. Complex.
• 2007
Let S={[email protected]?R^n|g"1(x)>=0,...,g"m(x)>=0} be a basic closed semialgebraic set defined by real polynomials g"i. Putinar's Positivstellensatz says that, under a certain condition stronger