Completely Positive Tensors and Multi-Hypergraphs

Completely positive graphs have been employed to associate with completely positive matrices for characterizing the intrinsic zero patterns. As tensors have been widely recognized as a higher-order extension of matrices, the multi-hypergraph, regarded as a generalization of graphs, is then introduced to associate with tensors for the study of complete positivity. To describe the dependence of the corresponding zero pattern for a special type of completely positive tensors--the $\{0,1\}$ completely positive tensors, the completely positive multi-hypergraph is defined. By characterizing properties of the associated multi-hypergraph, we provide necessary and sufficient conditions for any $(0,1)$ associated tensor to be $\{0,1\}$ completely positive. Furthermore, a necessary and sufficient condition for a uniform multi-hypergraph to be completely positive multi-hypergraph is proposed as well

Copositivity Detection of Tensors: Theory and Algorithm

A symmetric tensor is called copositive if it generates a multivariate form taking nonnegative values over the nonnegative orthant. Copositive tensors have found important applications in polynomial optimization and tensor complementarity problems. In this paper, we consider copositivity detection of tensors both from theoretical and computational points of view. After giving several necessary conditions for copositive tensors, we propose several new criteria for copositive tensors based on the representation of the multivariate form in barycentric coordinates with respect to the standard simplex and simplicial partitions. It is verified that, as the partition gets finer and finer, the concerned conditions eventually capture all strictly copositive tensors. Based on the obtained theoretical results with the help of simplicial partitions, we propose a numerical method to judge whether a tensor is copositive or not. The preliminary numerical results confirm our theoretical findings

Finding the maximum eigenvalue of a class of tensors with applications in copositivity test and hypergraphs

Finding the maximum eigenvalue of a symmetric tensor is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a semi-definite program algorithm for computing the maximum $H$-eigenvalue of a class of tensors with sign structure called $W$-tensors. The class of $W$-tensors extends the well-studied nonnegative tensors and essentially nonnegative tensors, and covers some important tensors arising naturally from spectral hypergraph theory. Our algorithm is based on a new structured sums-of-squares (SOS) decomposition result for a nonnegative homogeneous polynomial induced by a $W$-tensor. This SOS decomposition enables us to show that computing the maximum $H$-eigenvalue of an even order symmetric $W$-tensor is equivalent to solving a semi-definite program, and hence can be accomplished in polynomial time. Numerical examples are given to illustrate that the proposed algorithm can be used to find maximum $H$-eigenvalue of an even order symmetric $W$-tensor with dimension up to $10,000$. We present two applications for our proposed algorithm: we first provide a polynomial time algorithm for computing the maximum $H$-eigenvalues of large size Laplacian tensors of hyper-stars and hyper-trees; second, we show that the proposed SOS algorithm can be used to test the copositivity of a multivariate form associated with symmetric extended $Z$-tensors, whose order may be even or odd. Numerical experiments illustrate that our structured semi-definite program algorithm is effective and promising

An exact copositive programming formulation for the Discrete Ordered Median Problem: Extended version

This paper presents a first continuous, linear, conic formulation for the Discrete Ordered Median Problem (DOMP). Starting from a binary, quadratic formulation in the original space of location and allocation variables that are common in Location Analysis (L.A.), we prove that there exists a transformation of that formulation, using the same space of variables, that allows us to cast DOMP as a continuous linear problem in the space of completely positive matrices. This is the first positive result that states equivalence between the family of continuous convex problems and some hard problems in L.A. The result is of theoretical interest because it allows us to share the tools from continuous optimization to shed new light into the difficult combinatorial structure of the class of ordered median problems

Approximation Hierarchies for Copositive Tensor Cone

In this paper we discuss copositive tensors, which are a natural generalization of the copositive matrices. We present an analysis of some basic properties of copositive tensors; as well as the conditions under which class of copositive tensors and the class of positive semidefinite tensors coincides. Moreover, we have describe several hierarchies that approximates the cone of copositive tensors. The hierarchies are predominantly based on different regimes such as; simplicial partition, rational griding and polynomial conditions. The hierarchies approximates the copositive cone either from inside (inner approximation) or from outside (outer approximation). We will also discuss relationship among different hierarchies

Completely positive and completely positive semidefinite tensor relaxations for polynomial optimization

Completely positive (CP) tensors, which correspond to a generalization of CP matrices, allow to reformulate or approximate a general polynomial optimization problem (POP) with a conic optimization problem over the cone of CP tensors. Similarly, completely positive semidefinite (CPSD) tensors, which correspond to a generalization of positive semidefinite (PSD) matrices, can be used to approximate general POPs with a conic optimization problem over the cone of CPSD tensors. In this paper, we study CP and CPSD tensor relaxations for general POPs and compare them with the bounds obtained via a Lagrangian relaxation of the POPs. This shows that existing results in this direction for quadratic POPs extend to general POPs. Also, we provide some tractable approximation strategies for CP and CPSD tensor relaxations. These approximation strategies show that, with a similar computational effort, bounds obtained from them for general POPs can be tighter than bounds for these problems obtained by reformulating the POP as a quadratic POP, which subsequently can be approximated using CP and PSD matrices. To illustrate our results, we numerically compare the bounds obtained from these relaxation approaches on small scale fourth-order degree POPs

Positive polynomials on unbounded domains

Certificates of non-negativity such as Putinar's Positivstellensatz have been used to obtain powerful numerical techniques to solve polynomial optimization (PO) problems. Putinar's certificate uses sum-of-squares (sos) polynomials to certify the non-negativity of a given polynomial over a domain defined by polynomial inequalities. This certificate assumes the Archimedean property of the associated quadratic module, which in particular implies compactness of the domain. In this paper we characterize the existence of a certificate of non-negativity for polynomials over a possibly unbounded domain, without the use of the associated quadratic module. Next, we show that the certificate can be used to convergent linear matrix inequality (LMI) hierarchies for PO problems with unbounded feasible sets. Furthermore, by using copositive polynomials to certify non-negativity, instead of sos polynomials, the certificate allows the use of a very rich class of convergent LMI hierarchies to approximate the solution of general PO problems. Throughout the article we illustrate our results with various examples certifying the non-negativity of polynomials over possibly unbounded sets defined by polynomial equalities or inequalities

A Geometrical Analysis of a Class of Nonconvex Conic Programs for Convex Conic Reformulations of Quadratic and Polynomial Optimization Problems

We present a geometrical analysis on the completely positive programming reformulation of quadratic optimization problems and its extension to polynomial optimization problems with a class of geometrically defined nonconvex conic programs and their covexification. The class of nonconvex conic programs is described with a linear objective functionin a linear space $V$, and the constraint set is represented geometrically as the intersection of a nonconvex cone $K \subset V$, a face $J$ of the convex hull of $K$ and a parallel translation $L$ of a supporting hyperplane of the nonconvex cone $K$. We show that under a moderate assumption, the original nonconvex conic program can equivalently be reformulated as a convex conic program by replacing the constraint set with the intersection of $J$ and the hyperplane $L$. The replacement procedure is applied to derive the completely positive programming reformulation of quadratic optimization problems and its extension to polynomial optimization problems.Comment: 27 pages, 2 figure

Doubly Nonnegative Tensors, Completely Positive Tensors and Applications

The concept of double nonnegativity of matrices is generalized to doubly nonnegative tensors by means of the nonnegativity of all entries and $H$-eigenvalues. This generalization is defined for tensors of any order (even or odd), while it reduces to the class of nonnegative positive semidefinite tensors in the even order case. We show that many nonnegative structured tensors, which are positive semidefinite in the even order case, are indeed doubly nonnegative as well in the odd order case. As an important subclass of doubly nonnegative tensors, the completely positive tensors are further studied. By using dominance properties for completely positive tensors, we can easily exclude some doubly nonnegative tensors, such as the signless Laplacian tensor of a nonempty $m$-uniform hypergraph with $m\geq 3$, from the class of completely positive tensors. Properties of the doubly nonnegative tensor cone and the completely positive tensor cone are established. Their relation and difference are discussed. These show us a different phenomenon comparing to the matrix case. By employing the proposed properties, more subclasses of these two types of tensors are identified. Particularly, all positive Cauchy tensors with any order are shown to be completely positive. This gives an easily constructible subclass of completely positive tensors, which is significant for the study of completely positive tensor decomposition. A preprocessed Fan-Zhou algorithm is proposed which can efficiently verify the complete positivity of nonnegative symmetric tensors. We also give the solution analysis of tensor complementarity problems with the strongly doubly nonnegative tensor structure

The discrete moment problem with nonconvex shape constraints

The discrete moment problem is a foundational problem in distribution-free robust optimization, where the goal is to find a worst-case distribution that satisfies a given set of moments. This paper studies the discrete moment problems with additional "shape constraints" that guarantee the worst case distribution is either log-concave or has an increasing failure rate. These classes of shape constraints have not previously been studied in the literature, in part due to their inherent nonconvexities. Nonetheless, these classes of distributions are useful in practice. We characterize the structure of optimal extreme point distributions by developing new results in reverse convex optimization, a lesser-known tool previously employed in designing global optimization algorithms. We are able to show, for example, that an optimal extreme point solution to a moment problem with $m$ moments and log-concave shape constraints is piecewise geometric with at most $m$ pieces. Moreover, this structure allows us to design an exact algorithm for computing optimal solutions in a low-dimensional space of parameters. Moreover, We describe a computational approach to solving these low-dimensional problems, including numerical results for a representative set of instances.Comment: 31 pages, 3 figure