# 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} }

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## 9 Citations

### Complexity for exact polynomial optimization strengthened with Fritz John conditions

- Mathematics
- 2022

Let f, g 1 , . . . , g m be polynomials of degree at most d with real coeﬃcients in a vector of variables x = ( x 1 , . . . , x n ). Assume that f is non-negative on the basic semi-algebraic set S…

### Urysohn in action: separating semialgebraic sets by polynomials

- Mathematics, Computer Science
- 2022

The authors provide a decision algorithm for the more general separation problem without compactness assumptions and relies on positivity certificates based on sums of squares, such as Putinar certificates, for positive polynomials on basic compact semialgebraic sets.

### Sum-of-Squares Hierarchies for Polynomial Optimization and the Christoffel--Darboux Kernel

- Mathematics, Computer ScienceSIAM Journal on Optimization
- 2022

It is shown that the hierarchies based on the Schmüdgen-type certificates converge to the global minimum of f at a rate in O(1/r2), matching recently obtained convergence rates for the hypersphere and hypercube.

### Conic Linear Optimization for Computer-Assisted Proofs (hybrid meeting)

- Mathematics
- 2022

From a mathematical perspective, optimization is the science of proving inequalities. In this sense, computational optimization is a method for computer-assisted proofs. Conic (linear) optimization…

### On the effective Putinar’s Positivstellensatz and moment approximation

- MathematicsMathematical Programming
- 2022

We analyse the representation of positive polynomials in terms of Sums of Squares. We provide a quantitative version of Putinar’s Positivstellensatz over a compact basic semialgebraic set S , with a…

### Explicit degree bound for Nichtnegativstellens¨atze based on the Fritz John conditions

- Mathematics
- 2022

Let f, g 1 , . . . , g m be polynomials of degree at most d with real coeﬃcients in a vector of variables x = ( x 1 , . . . , x n ). Assume that f is nonnegative on the basic semi-algebraic set S…

### On Moment Approximation and the Effective Putinar’s Positivstellensatz

- Mathematics
- 2021

We analyse the representation of positive polynomials in terms of Sums of Squares. We provide a quantitative version of Putinar Positivstellensatz over a compact basic closed semialgebraic set S,…

### Exponential convergence of sum-of-squares hierarchies for trigonometric polynomials

- Mathematics, Computer Science
- 2022

We consider the unconstrained optimization of multivariate trigonometric polynomials by the sum-of-squares hierarchy of lower bounds. We ﬁrst show a convergence rate of O (1 /s 2 ) for the relaxation…

### Construction of multivariate polynomial approximation kernels via semidefinite programming

- Computer Science, Mathematics
- 2022

This paper constructs a hierarchy of multivariate polynomial approximation kernels via semideﬁnite pro- gramming and shows how a symmetry reduction may be performed to increase numerical tractability.

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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…

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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…

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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.

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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…