A survey of semidefinite programming approaches to the generalized problem of moments and their error analysis
17 Nov 2019-Vol. 20, Iss: 20, pp 17-56
TL;DR: This work will review some results on inner and inner conic approximations of the convex cone of positive Borel measures, with a special focus on the convergence rate of the hierarchies of upper and lower bounds for the general problem of moments that are obtained from these inner and outer approximation.
Abstract: 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, option 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 approximations, with a special focus on the convergence rate of the hierarchies of upper and lower bounds for the general problem of moments that are obtained from these inner and outer approximations.
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TL;DR: This analysis applies to simplices, balls and convex bodies that locally look like a ball, while also allowing for a broader class of reference measures, including the Lebesgue measure.
Abstract: We consider the problem of computing the minimum value fmin,K of a polynomial f over a compact set K⊆Rn, which can be reformulated as finding a probability measure ν on K minimizing ∫Kfdν. Lasserre showed that it suffices to consider such measures of the form ν=qμ, where q is a sum-of-squares polynomial and μ is a given Borel measure supported on K. By bounding the degree of q by 2r one gets a converging hierarchy of upper bounds f(r) for fmin,K. When K is the hypercube [−1,1]n, equipped with the Chebyshev measure, the parameters f(r) are known to converge to fmin,K at a rate in O(1/r2). We extend this error estimate to a wider class of convex bodies, while also allowing for a broader class of reference measures, including the Lebesgue measure. Our analysis applies to simplices, balls and convex bodies that locally look like a ball. In addition, we show an error estimate in O(logr/r) when K satisfies a minor geometrical condition, and in O(log2r/r2) when K is a convex body, equipped with the Lebesgue measure. This improves upon the currently best known error estimates in O(1/r√) and O(1/r) for these two respective cases.
31 citations
TL;DR: The convergence rate of a hierarchy of upper bounds for polynomial minimization problems, proposed by Lasserre (SIAM J Optim 21(3):864-885, 2011), for the special case when the feasible set is the unit (hyper)sphere was studied in this article.
Abstract: We study the convergence rate of a hierarchy of upper bounds for polynomial minimization problems, proposed by Lasserre (SIAM J Optim 21(3):864–885, 2011), for the special case when the feasible set is the unit (hyper)sphere. The upper bound at level r∈N of the hierarchy is defined as the minimal expected value of the polynomial over all probability distributions on the sphere, when the probability density function is a sum-of-squares polynomial of degree at most 2r with respect to the surface measure. We show that the rate of convergence is O(1/r2) and we give a class of polynomials of any positive degree for which this rate is tight. In addition, we explore the implications for the related rate of convergence for the generalized problem of moments on the sphere.
30 citations
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TL;DR: The rate of convergence of a hierarchy of upper bounds for polynomial minimization problems, proposed by Lasserre, is shown to be O(1/r^2) and a class of polynomials of any positive degree for which this rate is tight.
Abstract: We study the convergence rate of a hierarchy of upper bounds for polynomial minimization problems, proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp. 864-885], for the special case when the feasible set is the unit (hyper)sphere. The upper bound at level r of the hierarchy is defined as the minimal expected value of the polynomial over all probability distributions on the sphere, when the probability density function is a sum-of-squares polynomial of degree at most 2r with respect to the surface measure.
We show that the exact rate of convergence is Theta(1/r^2), and explore the implications for the related rate of convergence for the generalized problem of moments on the sphere.
19 citations
Cites background from "A survey of semidefinite programmin..."
...Ourmain result in Theorem 4 has the following implication for theGPMon the sphere, as a corollary of the following result in [12] (which applies to any compact K , see also [9] for a sketch of the proof in the setting described here)....
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Posted Content•
TL;DR: The proof combines classical Fourier analysis on $\mathbb{B}^{n}$ with the polynomial kernel technique and existing results on the extremal roots of Krawtchouk polynomials to establish the worst-case error of f_{(r) - f_{ (r)}, which is of the order of 1/2 - t(1-t)$ as $n$ tends to $\infty".
Abstract: We consider the sum-of-squares hierarchy of approximations for the problem of minimizing a polynomial $f$ over the boolean hypercube $\mathbb{B}^{n}=\{0,1\}^n$. This hierarchy provides for each integer $r \in \mathbb{N}$ a lower bound $f_{(r)}$ on the minimum $f_{\min}$ of $f$, given by the largest scalar $\lambda$ for which the polynomial $f - \lambda$ is a sum-of-squares on $\mathbb{B}^{n}$ with degree at most $2r$. We analyze the quality of these bounds by estimating the worst-case error $f_{\min} - f_{(r)}$ in terms of the least roots of the Krawtchouk polynomials. As a consequence, for fixed $t \in [0, 1/2]$, we can show that this worst-case error in the regime $r \approx t \cdot n$ is of the order $1/2 - \sqrt{t(1-t)}$ as $n$ tends to $\infty$. Our proof combines classical Fourier analysis on $\mathbb{B}^{n}$ with the polynomial kernel technique and existing results on the extremal roots of Krawtchouk polynomials. This link to roots of orthogonal polynomials relies on a connection between the hierarchy of lower bounds $f_{(r)}$ and another hierarchy of upper bounds $f^{(r)}$, for which we are also able to establish the same error analysis. Our analysis extends to the minimization of a polynomial over the $q$-ary cube $(\mathbb{Z}/q\mathbb{Z})^{n}$.
15 citations
TL;DR: 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.
Abstract: 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 ⊆ℝn. This hierarchy relies on using the push-forward measure of the Lebesgue measure on K by the polynomial f and involves univariate sums of squares of polynomials with growing degrees 2r. Hence it is weaker, but cheaper to compute, than an earlier hierarchy by Lasserre (SIAM Journal on Optimization 21(3), 864--885, 2011), which uses multivariate sums of squares. We show that this new hierarchy converges to the global minimum of f at a rate in O(log2 r / r2) whenever K satisfies a mild geometric condition, which holds, e.g., for convex bodies. As an application this rate of convergence also applies to the stronger hierarchy based on multivariate sums of squares, which extends earlier convergence results to a wider class of compact sets. Furthermore, we show that our analysis is near-optimal by proving a lower bound on the convergence rate in Ω(1/r2) for a class of polynomials on K=[-1,1], obtained by exploiting a connection to orthogonal polynomials.
8 citations
References
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TL;DR: 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.
Abstract: We consider the problem of finding the unconstrained global minimum of a real-valued polynomial p(x): {\mathbb{R}}^n\to {\mathbb{R}}$, as well as the global minimum of p(x), in a compact set K defined by polynomial inequalities. It is shown that this problem reduces to solving an (often finite) sequence of convex linear matrix inequality (LMI) problems. A notion of Karush--Kuhn--Tucker polynomials is introduced in a global optimality condition. Some illustrative examples are provided.
2,774 citations
01 Jun 2001
TL;DR: The authors present the basic theory of state-of-the-art polynomial time interior point methods for linear, conic quadratic, and semidefinite programming as well as their numerous applications in engineering.
Abstract: This is a book devoted to well-structured and thus efficiently solvable convex optimization problems, with emphasis on conic quadratic and semidefinite programming. The authors present the basic theory underlying these problems as well as their numerous applications in engineering, including synthesis of filters, Lyapunov stability analysis, and structural design. The authors also discuss the complexity issues and provide an overview of the basic theory of state-of-the-art polynomial time interior point methods for linear, conic quadratic, and semidefinite programming. The book's focus on well-structured convex problems in conic form allows for unified theoretical and algorithmical treatment of a wide spectrum of important optimization problems arising in applications.
2,651 citations
1,113 citations
"A survey of semidefinite programmin..." refers background in this paper
...We refer to [1] for an early reference and to the recent monograph [52] for a comprehensive treatment of the moment problem....
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..., [1, 31, 52] and references therein and [37] for a recent overview of many of its applications....
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Book•
02 Oct 2009TL;DR: The semidefinite programming methodology to solve the generalized problem of moments is presented and several applications of the GPM are described in detail (notably in optimization, probability, optimal control andmathematical finance).
Abstract: . From a theoretical viewpoint, the GPM has developments and impact in var-ious area of Mathematics like algebra, Fourier analysis, functional analysis, operator theory, probabilityand statistics, to cite a few. In addition, and despite its rather simple and short formulation, the GPMhas a large number of important applications in various fields like optimization, probability, mathematicalfinance, optimal control, control and signal processing, chemistry, cristallography, tomography, quantumcomputing, etc.In its full generality, the GPM is untractable numerically. However when K is a compact basic semi-algebraic set, and the functions involved are polynomials (and in some cases piecewise polynomials orrational functions), then the situation is much nicer. Indeed, one can define a systematic numerical schemebased on a hierarchy of semidefinite programs, which provides a monotone sequence that converges tothe optimal value of the GPM. (A semidefinite program is a convex optimization problem which up toarbitrary fixed precision, can be solved in polynomial time.) Sometimes finite convergence may evenocccur.In the talk, we will present the semidefinite programming methodology to solve the GPM and describein detail several applications of the GPM (notably in optimization, probability, optimal control andmathematical finance).R´ef´erences[1] J.B. Lasserre, Moments, Positive Polynomials and their Applications, Imperial College Press, inpress.[2] J.B. Lasserre, A Semidefinite programming approach to the generalized problem of moments,Math. Prog. 112 (2008), pp. 65–92.
1,020 citations
Book•
01 Jan 2002TL;DR: Convex sets at large Faces and extreme points ConveX sets in topological vector spaces Polarity, duality and linear programming Convex bodies and ellipsoids Faces of polytopes Lattices and convex bodies Lattice points and polyhedra.
Abstract: Convex sets at large Faces and extreme points Convex sets in topological vector spaces Polarity, duality and linear programming Convex bodies and ellipsoids Faces of polytopes Lattices and convex bodies Lattice points and polyhedra Bibliography Index.
861 citations