Convergence analysis for Lasserre’s measure-based hierarchy of upper bounds for polynomial optimization

@article{Klerk2017ConvergenceAF,
  title={Convergence analysis for Lasserre’s measure-based hierarchy of upper bounds for polynomial optimization},
  author={Etienne de Klerk and Monique Laurent and Zhao Sun},
  journal={Mathematical Programming},
  year={2017},
  volume={162},
  pages={363-392}
}
We consider the problem of minimizing a continuous function f over a compact set $${\mathbf {K}}$$K. We analyze a hierarchy of upper bounds proposed by Lasserre (SIAM J Optim 21(3):864–885, 2011), obtained by searching for an optimal probability density function h on $${\mathbf {K}}$$K which is a sum of squares of polynomials, so that the expectation $$\int _{{\mathbf {K}}} f(x)h(x)dx$$∫Kf(x)h(x)dx is minimized. We show that the rate of convergence is no worse than $$O(1/\sqrt{r})$$O(1/r… 
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References

SHOWING 1-10 OF 40 REFERENCES
A New Look at Nonnegativity on Closed Sets and Polynomial Optimization
TLDR
A convergent explicit hierarchy of semidefinite (outer) approximations with {\it no} lifting is obtained, of the cone of nonnegative polynomials of degree at most $d$, used in polynomial optimization on certain simple closed sets.
An Error Analysis for Polynomial Optimization over the Simplex Based on the Multivariate Hypergeometric Distribution
TLDR
This paper considers a rational approximation by taking the minimum over the regular grid, which consists of rational points with denominator $r$ (for given $r) and shows that the associated convergence rate is $O(1/r^2)$ for quadratic polynomials.
Unit balls of constant volume: which one has optimal representation?
In the family of unit balls with constant volume we look at the ones whose algebraic representation has some extremal property. We consider the family of nonnegative homogeneous polynomials of even
Certifying convergence of Lasserre’s hierarchy via flat truncation
TLDR
It is shown that flat truncation can be used as a certificate to check exactness of standard SOS relaxations and Jacobian SDP relaxations.
Global Optimization with Polynomials and the Problem of Moments
TLDR
It is shown that the problem of finding the unconstrained global minimum of a real-valued polynomial p(x): R n to R, in a compact set K defined byPolynomial inequalities reduces to solving an (often finite) sequence of convex linear matrix inequality (LMI) problems.
The condition number of real Vandermonde, Krylov and positive definite Hankel matrices
  • B. Beckermann
  • Mathematics, Computer Science
    Numerische Mathematik
  • 2000
Summary. We show that the Euclidean condition number of any positive definite Hankel matrix of order $n\geq 3$ may be bounded from below by $\gamma^{n-1}/(16n)$ with $\gamma=\exp(4 \cdot{\it
On the complexity of Putinar's Positivstellensatz
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
An alternative proof of a PTAS for fixed-degree polynomial optimization over the simplex
TLDR
New insight is provided into the polynomial-time approximation scheme for polynomials of fixed degree by establishing precise links with Bernstein approximation and the multinomial distribution.
Random walk in a simplex and quadratic optimization over convex polytopes
TLDR
Probabilistic arguments for justifying the quality of an approximate solution for global quadratic minimization problem are developed, obtained as a best point among all points of a uniform grid inside a polyhedral feasible set and some related problems are shown to be NP-hard.
...
...