Topic
Copositive matrix
About: Copositive matrix is a research topic. Over the lifetime, 54 publications have been published within this topic receiving 2370 citations.
Papers published on a yearly basis
Papers
More filters
TL;DR: A special class of indefinite quadratic programs is constructed, with simple constraints and integer data, and it is shown that checking (a) or (b) on this class is NP-complete.
Abstract: In continuous variable, smooth, nonconvex nonlinear programming, we analyze the complexity of checking whether(a)a given feasible solution is not a local minimum, and(b)the objective function is not bounded below on the set of feasible solutions.
We construct a special class of indefinite quadratic programs, with simple constraints and integer data, and show that checking (a) or (b) on this class is NP-complete. As a corollary, we show that checking whether a given integer square matrix is not copositive, is NP-complete.
1,117 citations
01 Jan 2010
TL;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 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
TL;DR: This work surveys essential properties of the so-called copositive matrices, the study of which has been spread over more than fifty-five years, with special emphasis on variational aspects related to the concept of copositivity.
Abstract: This work surveys essential properties of the so-called copositive matrices, the study of which has been spread over more than fifty-five years. Special emphasis is given to variational aspects related to the concept of copositivity. In addition, some new results on the geometry of the cone of copositive matrices are presented here for the first time.
157 citations
TL;DR: A strong duality relationship between the performance of a system and its stability analysis via mean drift is exhibited and this results carry over to fluid models, allowing the study of networks with nonexponential distributions.
Abstract: We consider the problems of performance analysis and stability/instability determination of queuing networks and scheduling policies. We exhibit a strong duality relationship between the performance of a system and its stability analysis via mean drift. We obtain a variety of linear programs (LPs) to conduct such stability and performance analyses. A certain LP, called the performance LP, bounds the performance of all stationary nonidling scheduling policies. If it is bounded, then its dual, called the drift LP, has a feasible solution which is a copositive matrix. The quadratic form associated with this copositive matrix has a negative drift, showing that all stationary nonidling scheduling policies result in a geometrically converging exponential moment. These results carry over to fluid models, allowing the study of networks with nonexponential distributions. If a modification of the performance LP, called the monotone LP, is bounded, then the system is stable. Finally, there is a another modification of the performance LP, called the finite time LP. It provides transient bounds on the performance of the system from any initial condition.
126 citations