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On Approximating Complex Quadratic Optimization Problems Via Semidefinite Programming Relaxations
 Mathematical Programming, Series B
, 2007
"... Abstract. In this paper we study semidefinite programming (SDP) models for a class of discrete and continuous quadratic optimization problems in the complex Hermitian form. These problems capture a class of well–known combinatorial optimization problems, as well as problems in control theory. For i ..."
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Abstract. In this paper we study semidefinite programming (SDP) models for a class of discrete and continuous quadratic optimization problems in the complex Hermitian form. These problems capture a class of well–known combinatorial optimization problems, as well as problems in control theory. For instance, they include Max–3–Cut with arbitrary edge weights (i.e. some of the edge weights might be negative). We present a generic algorithm and a unified analysis of the SDP relaxations which allow us to obtain good approximation guarantees for our models. Specifically, we give an (k sin(pi/k))2/(4pi)–approximation algorithm for the discrete problem where the decision variables are k– ary and the objective matrix is positive semidefinite. To the best of our knowledge, this is the first known approximation result for this family of problems. For the continuous problem where the objective matrix is positive semidefinite, we obtain the well–known pi/4 result due to [2], and independently, [12]. However, our techniques simplify their analyses and provide a unified framework for treating these problems. In addition, we show for the first time that the integrality gap of the SDP relaxation is precisely pi/4. We also show that the unified analysis can be used to obtain an O(1 / log n)–approximation algorithm for the continuous problem in the case where the objective matrix is not positive semidefinite. 1
Computable representations for convex hulls of lowdimensional quadratic forms
, 2007
"... Let C be the convex hull of points { ( 1) ( 1 x x)T  x ∈ F ⊂ ℜ n}. Representing or approximating C is a fundamental problem for global optimization algorithms based on convex relaxations of products of variables. If n ≤ 4 and F is a simplex then C has a computable representation in terms of matric ..."
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Cited by 30 (10 self)
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Let C be the convex hull of points { ( 1) ( 1 x x)T  x ∈ F ⊂ ℜ n}. Representing or approximating C is a fundamental problem for global optimization algorithms based on convex relaxations of products of variables. If n ≤ 4 and F is a simplex then C has a computable representation in terms of matrices X that are doubly nonnegative (positive semidefinite and componentwise nonnegative). If n = 2 and F is a box, then C has a representation that combines semidefiniteness with constraints on product terms obtained from the reformulationlinearization technique (RLT). The simplex result generalizes known representations for the convex hull of {(x1, x2, x1x2)  x ∈ F} when F ⊂ ℜ 2 is a triangle, while the result for box constraints generalizes the wellknown fact that in this case the RLT constraints generate the convex hull of {(x1, x2, x1x2)  x ∈ F}. When n = 3 and F is a box, a representation for C can be obtained by utilizing the simplex result for n = 4 in conjunction with a triangulation of the 3cube.
Semidefinite programming versus the reformulationlinearization technique for nonconvex quadratically constrained quadratic programming
, 2007
"... We consider relaxations for nonconvex quadratically constrained quadratic programming (QCQP) based on semidefinite programming (SDP) and the reformulationlinearization technique (RLT). From a theoretical standpoint we show that the addition of a semidefiniteness condition removes a substantial por ..."
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Cited by 28 (5 self)
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We consider relaxations for nonconvex quadratically constrained quadratic programming (QCQP) based on semidefinite programming (SDP) and the reformulationlinearization technique (RLT). From a theoretical standpoint we show that the addition of a semidefiniteness condition removes a substantial portion of the feasible region corresponding to product terms in the RLT relaxation. On test problems we show that the use of SDP and RLT constraints together can produce bounds that are substantially better than either technique used alone. For highly symmetric problems we also consider the effect of symmetrybreaking based on tightened bounds on variables and/or order constraints. 1 1
Quadratically constrained quadratic programs on acyclic graphs with application to power flow
, 2013
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Discretization and Localization in Successive Convex Relaxation Methods for Nonconvex Quadratic Optimization
, 2000
"... . Based on the authors' previous work which established theoretical foundations of two, conceptual, successive convex relaxation methods, i.e., the SSDP (Successive Semidefinite Programming) Relaxation Method and the SSILP (Successive SemiInfinite Linear Programming) Relaxation Method, this pa ..."
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Cited by 26 (13 self)
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. Based on the authors' previous work which established theoretical foundations of two, conceptual, successive convex relaxation methods, i.e., the SSDP (Successive Semidefinite Programming) Relaxation Method and the SSILP (Successive SemiInfinite Linear Programming) Relaxation Method, this paper proposes their implementable variants for general quadratic optimization problems. These problems have a linear objective function c T x to be maximized over a nonconvex compact feasible region F described by a finite number of quadratic inequalities. We introduce two new techniques, "discretization" and "localization," into the SSDP and SSILP Relaxation Methods. The discretization technique makes it possible to approximate an infinite number of semiinfinite SDPs (or semiinfinite LPs) which appeared at each iteration of the original methods by a finite number of standard SDPs (or standard LPs) with a finite number of linear inequality constraints. We establish: ffl Given any open convex ...
Second Order Cone Programming Relaxation of Nonconvex Quadratic Optimization Problems
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Approximation algorithms for homogeneous polynomial optimization with quadratic constraints
, 2009
"... In this paper, we consider approximation algorithms for optimizing a generic multivariate homogeneous polynomial function, subject to homogeneous quadratic constraints. Such optimization models have wide applications, e.g., in signal processing, magnetic resonance imaging (MRI), data training, appr ..."
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Cited by 25 (11 self)
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In this paper, we consider approximation algorithms for optimizing a generic multivariate homogeneous polynomial function, subject to homogeneous quadratic constraints. Such optimization models have wide applications, e.g., in signal processing, magnetic resonance imaging (MRI), data training, approximation theory, and portfolio selection. Since polynomial functions are nonconvex in general, the problems under consideration are all NPhard. In this paper we shall focus on polynomialtime approximation algorithms. In particular, we first study optimization of a multilinear tensor function over the Cartesian product of spheres. We shall propose approximation algorithms for such problem and derive worstcase performance ratios, which are shown to depend only on the dimensions of the models. The methods are then extended to optimize a generic multivariate homogeneous polynomial function with spherical constraints. Likewise, approximation algorithms are proposed with provable relative approximation performance ratios. Furthermore, the constraint set is relaxed to be an intersection of cocentered ellipsoids. In particular, we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin, and propose polynomialtime approximation algorithms with provable worstcase performance ratios. Numerical results are reported, illustrating the effectiveness of the approximation algorithms studied.
Advances in convex optimization: Conic programming
 In Proceedings of International Congress of Mathematicians
, 2007
"... Abstract. During the last two decades, major developments in convex optimization were focusing on conic programming, primarily, on linear, conic quadratic and semidefinite optimization. Conic programming allows to reveal rich structure which usually is possessed by a convex program and to exploit ..."
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Cited by 23 (0 self)
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Abstract. During the last two decades, major developments in convex optimization were focusing on conic programming, primarily, on linear, conic quadratic and semidefinite optimization. Conic programming allows to reveal rich structure which usually is possessed by a convex program and to exploit this structure in order to process the program efficiently. In the paper, we overview the major components of the resulting theory (conic duality and primaldual interior point polynomial time algorithms), outline the extremely rich “expressive abilities ” of conic quadratic and semidefinite programming and discuss a number of instructive applications.
Copositive Relaxation for General Quadratic Programming
 OPTIM. METHODS SOFTW
, 1998
"... We consider general, typically nonconvex, Quadratic Programming Problems. The Semidefinite relaxation proposed by Shor provides bounds on the optimal solution, but it does not always provide sufficiently strong bounds if linear constraints are also involved. To get rid of the linear sideconstraint ..."
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Cited by 22 (2 self)
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We consider general, typically nonconvex, Quadratic Programming Problems. The Semidefinite relaxation proposed by Shor provides bounds on the optimal solution, but it does not always provide sufficiently strong bounds if linear constraints are also involved. To get rid of the linear sideconstraints, another, stronger convex relaxation is derived. This relaxation uses copositive matrices. Special cases are discussed for which both relaxations are equal. At the end of the paper, the complexity and solvability of the relaxations are discussed.