New results on the cp-rank and related properties of co(mpletely )positive matrices
TL;DR: In this paper, the authors established new relations between orthogonal pairs of such matrices lying on the boundary of either cone and established an improvement on the upper bound of the cp-rank of completely positive matrices of general order.
Abstract: Copositive and completely positive matrices play an increasingly important role in Applied Mathematics, namely as a key concept for approximating NP-hard optimization problems. The cone of copositive matrices of a given order and the cone of completely positive matrices of the same order are dual to each other with respect to the standard scalar product on the space of symmetric matrices. This paper establishes some new relations between orthogonal pairs of such matrices lying on the boundary of either cone. As a consequence, we can establish an improvement on the upper bound of the cp-rank of completely positive matrices of general order and a further improvement for such matrices of order six.
Citations
More filters
••
TL;DR: The main open problems in the theory of copositive and completely positive matrices are described in this paper, with a brief description of the state of the art in each open problem.
Abstract: We describe the main open problems which are currently of interest in the theory of copositive and completely positive matrices. We give motivation as to why these questions are relevant and provide a brief description of the state of the art in each open problem.
67 citations
••
TL;DR: It is shown that two-stage robust and distributionally robust linear programs can often be reformulated exactly as conic programs that scale polynomially with the problem dimensions.
Abstract: Adaptive robust optimization problems are usually solved approximately by restricting the adaptive decisions to simple parametric decision rules. However, the corresponding approximation error can be substantial. In this paper we show that two-stage robust and distributionally robust linear programs can often be reformulated exactly as conic programs that scale polynomially with the problem dimensions. Specifically, when the ambiguity set constitutes a 2-Wasserstein ball centered at a discrete distribution, then the distributionally robust linear program is equivalent to a copositive program (if the problem has complete recourse) or can be approximated arbitrarily closely by a sequence of copositive programs (if the problem has sufficiently expensive recourse). These results directly extend to the classical robust setting and motivate strong tractable approximations of two-stage problems based on semidefinite approximations of the copositive cone. We also demonstrate that the two-stage distributionally robust optimization problem is equivalent to a tractable linear program when the ambiguity set constitutes a 1-Wasserstein ball centered at a discrete distribution and there are no support constraints.
64 citations
••
TL;DR: This paper constructs counterexamples to the DJL conjecture for all $n\ge {12}$ and shows the largest possible cp-rank of an completely positive matrix, p_n, is defined.
Abstract: Let $p_n$ denote the largest possible cp-rank of an $n\times n$ completely positive matrix. This matrix parameter has its significance both in theory and applications, as it sheds light on the geometry and structure of the solution set of hard optimization problems in their completely positive formulation. Known bounds for $p_n$ are $s_n=\binom{n+1}2-4$, the current best upper bound, and the Drew--Johnson--Loewy (DJL) lower bound $d_n=\lfloor\frac{n^2}4\rfloor$. The famous DJL conjecture (1994) states that $p_n=d_n$. Here we show $p_n=\frac {n^2}2 +{\mathcal O}(n^{3/2}) = 2d_n+{\mathcal O}(n^{3/2}){{p_n=\frac {n^2}2 +{\mathcal O}(n^{3/2}) = 2d_n+{\mathcal O}(n^{3/2})}}$, and construct counterexamples to the DJL conjecture for all $n\ge {12}$ (for orders seven through eleven counterexamples were already given in [I. M. Bomze, W. Schachinger, and R. Ullrich, Linear Algebra Appl., 459 (2014), pp. 208--221].
43 citations
••
TL;DR: In this article, the authors studied n × n completely positive matrices M on the boundary of the completely positive cone, namely those orthogonal to a copositive matrix S which generates a quadratic form with finitely many zeroes in the standard simplex.
36 citations
••
TL;DR: A class of completely positive matrices with quadratic (in terms of the matrix size) completely positive rank, but with linear completely positive semidefinite rank is exhibited, and a connection to the existence of Hadamard matrices is made.
28 citations
References
More filters
•
15 Apr 2003TL;DR: In this article, the PSD Completion Problem Complete Positivity: Definition and Basic Properties Cones of Completely Positive Matrices Small Matrices complete positive matrix Small matrices complete positivity and the comparison matrix Completely positive graphs complete positive graph matrix complete positive graphs complete positive matrices of a given rank Complete positive matrix of the graph.
Abstract: Matrix Theoretic Background Positive Semidefinite Matrices Nonnegative Matrices and M-Matrices Schur Complements Graphs Convex Cones The PSD Completion Problem Complete Positivity: Definition and Basic Properties Cones of Completely Positive Matrices Small Matrices Complete Positivity and the Comparison Matrix Completely Positive Graphs Completely Positive Matrices Whose Graphs are Not Completely Positive Square Factorizations Functions of Completely Positive Matrices The CP Completion Problem CP Rank: Definition and Basic Results Completely Positive Matrices of a Given Rank Completely Positive Matrices of a Given Order When is the CP-Rank Equal to the Rank?
336 citations
••
01 Jan 2010TL;DR: The purpose of this survey is to introduce the field to interested readers in the optimization community who wish to get an understanding of the basic concepts and recent developments inCopositive programming, including modeling issues and applications, the connection to semidefinite programming and sum-of-squares approaches, as well as algorithmic solution approaches for copositive programs.
Abstract: Copositive programming is a relatively young field in mathematical optimization. It can be seen as a generalization of semidefinite programming, since it means optimizing over the cone of so called copositive matrices. Like semidefinite programming, it has proved particularly useful in combinatorial and quadratic optimization. The purpose of this survey is to introduce the field to interested readers in the optimization community who wish to get an understanding of the basic concepts and recent developments in copositive programming, including modeling issues and applications, the connection to semidefinite programming and sum-of-squares approaches, as well as algorithmic solution approaches for copositive programs.
255 citations
••
01 Jan 1962TL;DR: In this article, the problem of finding the constants a1, …, an so that xn+i = xi ≥ 0 for all i is solved. But the solution is not complete.
Abstract: In a recent paper (3) Mordell proposed for solutionPROBLEM 1. To find the constants a1, …, an so thatwhere xn+i = xi ≥ 0 for all i.
184 citations
••
TL;DR: Copositivity appears in local and global optimality conditions for quadratic optimization, but can also yield tighter bounds for NP-hard combinatorial optimization problems.
163 citations
••
01 Apr 1963
TL;DR: A completely positive quadratic form is a real form which can be written as a sum of squares of non-negative real forms as mentioned in this paper, which is the basic block designs arising in combinatorial analysis.
Abstract: A copositive quadratic form is a real form which is non-negative for non-negative arguments. A completely positive quadratic form is a real form which can be written as a sum of squares of non-negative real forms. The completely positive forms are basic in the study of block designs arising in combinatorial analysis (3). The copositive forms arise in the theory of inequalities and have been considered in a paper by Mordell (4) and two papers by Diananda(1, 2).
159 citations