This paper is concerned with minimizing a sum of rational functions over a compact set of high-dimension. Our approach relies on the second Lasserre's hierarchy (also known as the upper bounds hierarchy) formulated on the pushforward measure in order to work in a space of smaller dimension. We show that in the general case the minimum can be approximated as closely as desired from above with a hierarchy of semidefinite programs problems or, in the particular case of a single fraction, with a hierarchy of generalized eigenvalue problems.
We numerically illustrate the potential of using the pushforward measure rather than the standard upper bounds hierarchy. In our opinion, this potential should be a strong incentive to investigate a related challenging problem interesting in its own; namely integrating an arbitrary power of a given polynomial on a simple set (e.g., unit box or unit sphere) with respect to Lebesgue or Haar measure.
A polynomial optimization problem (POP) consists of minimizing a multivariate real polynomial on a semi-algebraic set $K$ described by polynomial inequalities and equations. In its full generality it is a non-convex, multi-extremal, difficult global optimization problem. More than an decade ago, J.~B.~Lasserre proposed to solve POPs by a hierarchy of convex semidefinite programming (SDP) relaxations of increasing size. Each problem in the hierarchy has a primal SDP formulation (a relaxation of a moment problem) and a dual SDP formulation (a sum-of-squares representation of a polynomial Lagrang…
We consider the problem of minimizing a continuous function f over a compact set K. We analyze a hierarchy of upper bounds proposed by Lasserre in [SIAM J. Optim. 21(3) (2011), pp. 864--885], obtained by searching for an optimal probability density function h on K which is a sum of squares of polynomials, so that the expectation $\int_{K}f(x)h(x)dx$ 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 K is a full-dimensional compact set sati…
We apply methods and techniques of tropical optimization to develop a new theoretical and computational framework for the implementation of the Analytic Hierarchy Process in multi-criteria problems of rating alternatives from pairwise comparison data. In this framework, we first consider the minimax Chebyshev approximation of pairwise comparison matrices by consistent matrices in the logarithmic scale. Recasting this approximation problem as a problem of tropical pseudo-quadratic programming we then write out a closed-form solution to it. This solution might be either a unique score vector (up…
Comparison of Lasserre's measure--based bounds for polynomial optimization to bounds obtained by simulated annealing. We consider the problem of minimizing a continuous function $f$ over a compact set $\mathbf{K}$. We compare the hierarchy of upper bounds proposed by Lasserre in [{\em SIAM J. Optim.} $21(3)$ $(2011)$, pp. $864-885$] to bounds that may be obtained from simulated annealing. We show that, when $f$ is a polynomial and $\mathbf{K}$ a convex body, this comparison yields a faster rate of convergence of the Lasserre hierarchy than what was previously known in the literature.
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 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 $λ$ for which the polynomial $f - λ$ 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 […
Mines Albi（阿尔比矿业学院）；Institut Clément Ader (ICA)（克莱门特·阿登研究所）；CNRS（法国国家科学研究中心；图卢兹实验室分析与系统研究）；LAAS（图卢兹大学；图卢兹第三大学；国立应用科学学院；国立高等航空航天学院；图卢兹实验室分析与系统研究）；Université de Toulouse（捷克布拉格捷克理工大学电气工程学院）；UPS, INSA, INP, ISAE（图卢兹大学图卢兹数学研究所）；LAAS；Faculty of Electrical Engineering, Czech Technical University in Prague；Institut de Mathématiques de Toulouse (IMT), Université de Toulouse
Global weight optimization of frame structures under free-vibration eigenvalue constraints
Czech Technical University in Prague（捷克布拉格理工大学）；Institute of Information Theory and Automation, Czech Academy of Sciences（捷克科学院信息理论与自动化研究所）；University of Birmingham（伯明翰大学）；Nečas Center for Mathematical Modeling（内卡什数学建模中心）
We consider the problem of computing the minimum value $f_{\min,K}$ of a polynomial $f$ over a compact set $K \subseteq \mathbb{R}^n$, which can be reformulated as finding a probability measure $ν$ on $K$ minimizing $\int_K f dν$. 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 $f_{\min,K}$. When $K$ is the hypercube $[-1, 1]^n$, equipped with the Chebyshev measure, the par…
Simple Approximations of Semialgebraic Sets and their Applications to Control
CNR-IEIIT（意大利国家研究委员会工业工程研究所）；Politecnico di Torino（都灵理工大学）；CNRS（法国国家科学研究中心）；LAAS（图卢兹系统分析与实验室）；Université de Toulouse（图卢兹大学）；Faculty of Electrical Engineering, Czech Technical University in Prague（布拉格捷克理工大学电气工程学院）；Electrical Engineering Department, The Pennsylvania State University（宾夕法尼亚州立大学电气工程系）
