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Semidefinite relaxation of quadratic optimization problems
 SIGNAL PROCESSING MAGAZINE, IEEE
, 2010
"... n recent years, the semidefinite relaxation (SDR) technique has been at the center of some of very exciting developments in the area of signal processing and communications, and it has shown great significance and relevance on a variety of applications. Roughly speaking, SDR is a powerful, computa ..."
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Cited by 150 (9 self)
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, computationally efficient approximation technique for a host of very difficult optimization problems. In particular, it can be applied to many nonconvex quadratically constrained quadratic programs (QCQPs) in an almost mechanical fashion, including the following problem: min x[Rn x T
Iteration Algorithm for Computing Bounds in Quadratic Optimization Problems
 Complexity in Numerical Optimization
, 1993
"... We consider the problem of optimizing a quadratic function subject to integer constraints. This problem is NPhard in the general case. We present a new polynomial time algorithm for computing bounds on the solutions to such optimization problems. We transform the problem into a problem for minimizi ..."
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Cited by 6 (0 self)
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We consider the problem of optimizing a quadratic function subject to integer constraints. This problem is NPhard in the general case. We present a new polynomial time algorithm for computing bounds on the solutions to such optimization problems. We transform the problem into a problem
Sparse Solutions to Random Standard Quadratic Optimization Problems
, 2011
"... The standard quadratic optimization problem (StQP) refers to the problem of minimizing a quadratic form over the standard simplex. Such a problem arises from numerous applications and is known to be NPhard. In this paper we focus on a special scenario of the StQP where all the elements of the data ..."
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The standard quadratic optimization problem (StQP) refers to the problem of minimizing a quadratic form over the standard simplex. Such a problem arises from numerous applications and is known to be NPhard. In this paper we focus on a special scenario of the StQP where all the elements of the data
On Copositive Programming and Standard Quadratic Optimization Problems
 Journal of Global Optimization
, 2000
"... A standard quadratic problem consists of finding global maximizers of a quadratic form over the standard simplex. In this paper, the usual semidefinite programming relaxation is strengthened by replacing the cone of positive semidefinite matrices by the cone of completely positive matrices (the posi ..."
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Cited by 41 (9 self)
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A standard quadratic problem consists of finding global maximizers of a quadratic form over the standard simplex. In this paper, the usual semidefinite programming relaxation is strengthened by replacing the cone of positive semidefinite matrices by the cone of completely positive matrices (the
Solving Standard Quadratic Optimization Problems Via Linear, Semidefinite and Copositive Programming
 J. GLOBAL OPTIM
, 2001
"... The problem of minimizing a (nonconvex) quadratic function over the simplex (the standard quadratic optimization problem) has an exact convex reformulation as a copositive programming problem. In this paper we show how to approximate the optimal solution by approximating the cone of copositive matr ..."
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Cited by 52 (6 self)
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The problem of minimizing a (nonconvex) quadratic function over the simplex (the standard quadratic optimization problem) has an exact convex reformulation as a copositive programming problem. In this paper we show how to approximate the optimal solution by approximating the cone of copositive
On Some Quadratic Optimization Problems Arising in Computer Vision
, 2014
"... Abstract. The goal of this paper is to find methods for solving various quadratic optimization problems, mostly arising from computer vision (image segmentation and contour grouping). We consider mainly two problems: Problem 1. Let A be an n × n Hermitian matrix and let b ∈ Cn be any vector, maximi ..."
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Abstract. The goal of this paper is to find methods for solving various quadratic optimization problems, mostly arising from computer vision (image segmentation and contour grouping). We consider mainly two problems: Problem 1. Let A be an n × n Hermitian matrix and let b ∈ Cn be any vector
On the Equivalence of Quadratic Optimization Problems Commonly Used in Portfolio Theory
"... In the paper, we consider three quadratic optimization problems which are frequently applied in portfolio theory, i.e, the Markowitz meanvariance problem as well as the problems based on the meanvariance utility function and the quadratic utility. Conditions are derived under which the solutions o ..."
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In the paper, we consider three quadratic optimization problems which are frequently applied in portfolio theory, i.e, the Markowitz meanvariance problem as well as the problems based on the meanvariance utility function and the quadratic utility. Conditions are derived under which the solutions
Towards Implementations of Successive Convex Relaxation Methods for Nonconvex Quadratic Optimization Problems
, 1999
"... Recently Kojima and Tuncel proposed new successive convex relaxation methods and their localizeddiscretized variants for general nonconvex quadratic optimization problems. Although an upper bound of the optimal objective function value within a previously given precision can be found theoretically ..."
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Cited by 11 (6 self)
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Recently Kojima and Tuncel proposed new successive convex relaxation methods and their localizeddiscretized variants for general nonconvex quadratic optimization problems. Although an upper bound of the optimal objective function value within a previously given precision can be found theoretically
SDP Relaxations for Quadratic Optimization Problems Derived from Polynomial Optimization Problems
, 2009
"... Based on the convergent sequence of SDP relaxations for a multivariate polynomial optimization problem (POP) by Lasserre, Waki et al. constructed a sequence of sparse SDP relaxations to solve sparse POPs efficiently. Nevertheless, the size of the sparse SDP relaxation is the major obstacle in order ..."
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to solve POPs of higher degree. This paper proposes an approach to transform general POPs to quadratic optimization problems (QOPs), which allows to reduce the size of the SDP relaxation substantially. We introduce different heuristics resulting in equivalent QOPs and show how sparsity of a POP
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