A Semidefinite Approach for Truncated K-Moment Problems

A truncated moment sequence (tms) of degree d is a vector indexed by monomials whose degree is at most d. Let K be a semialgebraic set.The truncated K-moment problem (TKMP) is: when does a tms y admit a positive Borel measure supported? This paper proposes a semidefinite programming (SDP) approach for solving TKMP. When K is compact, we get the following results: whether a tms y of degree d admits a K-measure or notcan be checked via solving a sequence of SDP problems; when y admits no K-measure, a certificate will be given; when y admits a K-measure, a representing measure for y would be obtained from solving the SDP under some necessary and some sufficient conditions. Moreover, we also propose a practical SDP method for finding flat extensions, which in our numerical experiments always finds a finitely atomic representing measure for a tms when it admits one

Computation of Real Radical Ideals by Semidefinite Programming and Iterative Methods

Systems of polynomial equations with approximate real coefficients arise frequently as models in applications in science and engineering. In the case of a system with finitely many real solutions (the $0$ dimensional case), an equivalent system generates the so-called real radical ideal of the system. In this case the equivalent real radical system has only real (i.e., no non-real) roots and no multiple roots. Such systems have obvious advantages in applications, including not having to deal with a potentially large number of non-physical complex roots, or with the ill-conditioning associated with roots with multiplicity. There is a corresponding, but more involved, description of the real radical for systems with real manifolds of solutions (the positive dimensional case) with corresponding advantages in applications. The stable and practical computation of real radicals in the approximate case is an important open problem. Theoretical advances and corresponding implemented algorithms are made for this problem. The approach of the thesis, is to use semidefinite programming (SDP) methods from algebraic geometry, and also techniques originating in the geometry of differential equations. The problem of finding the real radical is re-formulated as solving an SDP problem. This approach in the $0$ dimensional case, was pioneered by Curto \& Fialkow with breakthroughs in the $0$ dimensional case by Lasserre and collaborators. In the positive dimensional case, important contributions have been made of Ma, Wang and Zhi. The real radical corresponds to a generic point lying on the intersection of boundary of the convex cone of semidefinite matrices and a linear affine space associated with the polynomial system. As posed, this problem is not stable, since an arbitrarily small perturbation takes the point to an infeasible one outside the cone. A contribution of the thesis, is to show how to apply facial reduction pioneered by Borwein and Wolkowicz, to this problem. It is regularized by mapping the point to one which is strictly on the interior of another convex region, the minimal face of the cone. Then a strictly feasible point on the minimal face can be computed by accurate iterative methods such as the Douglas-Rachford method. Such a point corresponds to a generic point (max rank solution) of the SDP feasible problem. The regularization is done by solving the auxiliary problem which can be done again by iterative methods. This process is proved to be stable under some assumptions in this thesis as the max rank doesn\u27t change under sufficiently small perturbations. This well-posedness is also reflected in our examples, which are executed much more accurately than by methods based on interior point approaches. For a given polynomial system, and an integer $d \u3e 0$, Results of Curto \& Fialkow and Lasserre are generalized to give an algorithm for computing the real radical up to degree $d$. Using this truncated real radical as input to critical point methods, can lead in many cases to validation of the real radical

Computation with Polynomial Equations and Inequalities arising in Combinatorial Optimization

The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear algebra or semidefinite programming relaxations of many kinds of feasibility or optimization questions. We are particularly interested in problems arising in combinatorial optimization.Comment: 28 pages, survey pape

Polynomial systems : graphical structure, geometry, and applications

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2018.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (pages 199-208).Solving systems of polynomial equations is a foundational problem in computational mathematics, that has several applications in the sciences and engineering. A closely related problem, also prevalent in applications, is that of optimizing polynomial functions subject to polynomial constraints. In this thesis we propose novel methods for both of these tasks. By taking advantage of the graphical and geometrical structure of the problem, our methods can achieve higher efficiency, and we can also prove better guarantees. Various problems in areas such as robotics, power systems, computer vision, cryptography, and chemical reaction networks, can be modeled by systems of polynomial equations, and in many cases the resulting systems have a simple sparsity structure. In the first part of this thesis we represent this sparsity structure with a graph, and study the algorithmic and complexity consequences of this graphical abstraction. Our main contribution is the introduction of a novel data structure, chordal networks, that always preserves the underlying graphical structure of the system. Remarkably, many interesting families of polynomial systems admit compact chordal network representations (of size linear in the number of variables), even though the number of components is exponentially large. Our methods outperform existing techniques by orders of magnitude in applications from algebraic statistics and vector addition systems. We then turn our attention to the study of graphical structure in the computation of matrix permanents, a classical problem from computer science. We provide a novel algorithm that requires Õ(n 2[superscript w]) arithmetic operations, where [superscript w] is the treewidth of its bipartite adjacency graph. We also investigate the complexity of some related problems, including mixed discriminants, hyperdeterminants, and mixed volumes. Although seemingly unrelated to polynomial systems, our results have natural implications on the complexity of solving sparse systems. The second part of this thesis focuses on the problem of minimizing a polynomial function subject to polynomial equality constraints. This problem captures many important applications, including Max-Cut, tensor low rank approximation, the triangulation problem, and rotation synchronization. Although these problems are nonconvex, tractable semidefinite programming (SDP) relaxations have been proposed. We introduce a methodology to derive more efficient (smaller) relaxations, by leveraging the geometrical structure of the underlying variety. The main idea behind our method is to describe the variety with a generic set of samples, instead of relying on an algebraic description. Our methods are particularly appealing for varieties that are easy to sample from, such as SO(n), Grassmannians, or rank k tensors. For arbitrary varieties we can take advantage of the tools from numerical algebraic geometry. Optimization problems from applications usually involve parameters (e.g., the data), and there is often a natural value of the parameters for which SDP relaxations solve the (polynomial) problem exactly. The final contribution of this thesis is to establish sufficient conditions (and quantitative bounds) under which SDP relaxations will continue to be exact as the parameter moves in a neighborhood of the original one. Our results can be used to show that several statistical estimation problems are solved exactly by SDP relaxations in the low noise regime. In particular, we prove this for the triangulation problem, rotation synchronization, rank one tensor approximation, and weighted orthogonal Procrustes.by Diego Cifuentes.Ph. D