Abstract
We consider the problem of minimizing a continuous function f over a compact set \({\mathbf {K}}\). 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 \({\mathbf {K}}\) which is a sum of squares of polynomials, so that the expectation \(\int _{{\mathbf {K}}} f(x)h(x)dx\) is minimized. We show that the rate of convergence is no worse than \(O(1/\sqrt{r})\), where 2r is the degree bound on the density function. This analysis applies to the case when f is Lipschitz continuous and \({\mathbf {K}}\) is a full-dimensional compact set satisfying some boundary condition (which is satisfied, e.g., for convex bodies). The rth upper bound in the hierarchy may be computed using semidefinite programming if f is a polynomial of degree d, and if all moments of order up to \(2r+d\) of the Lebesgue measure on \({\mathbf {K}}\) are known, which holds, for example, if \({\mathbf {K}}\) is a simplex, hypercube, or a Euclidean ball.
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Acknowledgments
We thank Jean Bernard Lasserre for bringing our attention to his work [18] and for several valuable suggestions, and Dorota Kurowicka for valuable discussions on multivariate sampling techniques.
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de Klerk, E., Laurent, M. & Sun, Z. Convergence analysis for Lasserre’s measure-based hierarchy of upper bounds for polynomial optimization. Math. Program. 162, 363–392 (2017). https://doi.org/10.1007/s10107-016-1043-1
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DOI: https://doi.org/10.1007/s10107-016-1043-1