Improved convergence rates for Lasserre-type hierarchies of upper bounds for box-constrained polynomial optimization
Etienne de Klerk, Roxana Hess, Monique Laurent
备注：
21 pages, 3 figures, published in SIAM Journal on Optimization, 27(1):347-367, March 2017. This arXiv version is an updated version of the journal paper. Typos were corrected in the proofs of Lemmata 2.2 and 3.5 and Theorem 4.2
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 polynomial density function $h$ (of given degree $r$) that minimizes the expectation $\int_{[-1,1]^n} f(x)h(x)dμ(x)$, where $dμ(x)$ is a fixed, finite Borel measure supported on $[-1,1]^n$. It is known that, for the Lebesgue measure $dμ(x) = dx$, one may show an error bound $O(1/\sqrt{r})$ if $h$ is a sum-of-squares density, and an $O(1/r)$ error bound if $h$ is the density of a beta distribution. In this paper, we show an error bound of $O(1/r^2)$, if $dμ(x) = \left(\prod_{i=1}^n \sqrt{1-x_i^2} \right)^{-1}dx$ (the well-known measure in the study of orthogonal polynomials), and $h$ has a Schmüdgen-type representation with respect to $[-1,1]^n$, which is a more general condition than a sum of squares. The convergence rate analysis relies on the theory of polynomial kernels, and in particular on Jackson kernels. We also show that the resulting upper bounds may be computed as generalized eigenvalue problems, as is also the case for sum-of-squares densities.
We consider the generalized moment problem (GMP) over the simplex and the sphere. This is a rich setting and it contains NP-hard problems as special cases, like constructing optimal cubature schemes and rational optimization. Using the Reformulation-Linearization Technique (RLT) and Lasserre-type hierarchies, relaxations of the problem are introduced and analyzed. For our analysis we assume throughout the existence of a dual optimal solution as well as strong duality. For the GMP over the simplex we prove a convergence rate of $O(1/r)$ for a linear programming, RLT-type hierarchy, where $r$ is…
We exhibit a convex polynomial optimization problem for which the diagonally-dominant sum-of-squares (DSOS) and the scaled diagonally-dominant sum-of-squares (SDSOS) hierarchies, based on linear programming and second-order conic programming respectively, do not converge to the global infimum. The same goes for the r-DSOS and r-SDSOS hierarchies. This refutes the claim in the literature according to which the DSOS and SDSOS hierarchies can solve any polynomial optimization problem to arbitrary accuracy. In contrast, the Lasserre hierarchy based on semidefinite programming yields the global inf…
We study the convergence rate of a hierarchy of upper bounds for polynomial optimization problems, proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp. 864-885], and a related hierarchy by De Klerk, Hess and Laurent [SIAM J. Optim. 27(1), (2017) pp. 347-367]. For polynomial optimization over the hypercube, we show a refined convergence analysis for the first hierarchy. We also show lower bounds on the convergence rate for both hierarchies on a class of examples. These lower bounds match the upper bounds and thus establish the true rate of convergence on these examples. Interestingly, these c…
Lovasz and Schrijver, and later Lasserre, proposed hierarchies of semidefinite programming relaxations for general 0/1 linear programming problems. In this paper these two constructions are revisited and two new, block-diagonal hierarchies are proposed. They have the advantage of being computationally less costly while being at least as strong as the Lovasz-Schrijver hierarchy. Our construction is applied to the stable set problem and experimental results for Paley graphs are reported.
We consider the problem of the semidefinite representation of a class of non-compact basic semialgebraic sets. We introduce the conditions of pointedness and closedness at infinity of a semialgebraic set and show that under these conditions our modified hierarchies of nested theta bodies and Lasserre's relaxations converge to the closure of the convex hull of $S$. Moreover, if the PP-BDR property is satisfied, our theta body and Lasserre's relaxation are exact when the order is large enough; if the PP-BDR property does not hold, our hierarchies convergent uniformly to the closure of the convex…
We consider the problem of approximating numerically the moments and the supports of measures which are invariant with respect to the dynamics of continuous- and discrete-time polynomial systems, under semialgebraic set constraints. First, we address the problem of approximating the density and hence the support of an invariant measure which is absolutely continuous with respect to the Lebesgue measure. Then, we focus on the approximation of the support of an invariant measure which is singular with respect to the Lebesgue measure. Each problem is handled through an appropriate reformulation i…
SPLD polynomial optimization and bounded degree SOS hierarchies
Academy for Advanced Interdisciplinary Studies, Northeast Normal University（东北师范大学前沿交叉学科学院）；Shanghai Zhangjiang Institute of Mathematics（上海张江数学研究院）；Department of Applied Mathematics, Pukyong National University（釜庆国立大学应用数学系）；Department of Pedagogy, Dalat University（大叻大学教育学系）
The generalized problem of moments is a conic linear optimization problem over the convex cone of positive Borel measures with given support. It has a large variety of applications, including global optimization of polynomials and rational functions, options pricing in finance, constructing quadrature schemes for numerical integration, and distributionally robust optimization. A usual solution approach, due to J.B. Lasserre, is to approximate the convex cone of positive Borel measures by finite dimensional outer and inner conic approximations. We will review some results on these approximation…
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