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37
Approximation bounds for quadratic optimization with homogeneous quadratic constraints
 SIAM J. Optim
, 2007
"... Abstract. We consider the NPhard problem of finding a minimum norm vector in ndimensional real or complex Euclidean space, subject to m concave homogeneous quadratic constraints. We show that a semidefinite programming (SDP) relaxation for this nonconvex quadratically constrained quadratic program ..."
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Cited by 49 (24 self)
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Abstract. We consider the NPhard problem of finding a minimum norm vector in ndimensional real or complex Euclidean space, subject to m concave homogeneous quadratic constraints. We show that a semidefinite programming (SDP) relaxation for this nonconvex quadratically constrained quadratic program (QP) provides an O(m 2) approximation in the real case and an O(m) approximation in the complex case. Moreover, we show that these bounds are tight up to a constant factor. When the Hessian of each constraint function is of rank 1 (namely, outer products of some given socalled steering vectors) and the phase spread of the entries of these steering vectors are bounded away from π/2, we establish a certain “constant factor ” approximation (depending on the phase spread but independent of m and n) for both the SDP relaxation and a convex QP restriction of the original NPhard problem. Finally, we consider a related problem of finding a maximum norm vector subject to m convex homogeneous quadratic constraints. We show that an SDP relaxation for this nonconvex QP provides an O(1 / ln(m)) approximation, which is analogous to a result of Nemirovski et al. [Math. Program., 86 (1999), pp. 463–473] for the real case. Key words. semidefinite programming relaxation, nonconvex quadratic optimization, approximation bound
Spectrum Sharing in Wireless Networks via QoSAware Secondary Multicast Beamforming
"... Abstract—Secondary spectrum usage has the potential to considerably increase spectrum utilization. In this paper, qualityofservice (QoS)aware spectrum underlay of a secondary multicast network is considered. A multiantenna secondary access point (AP) is used for multicast (common information) tra ..."
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Cited by 40 (16 self)
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Abstract—Secondary spectrum usage has the potential to considerably increase spectrum utilization. In this paper, qualityofservice (QoS)aware spectrum underlay of a secondary multicast network is considered. A multiantenna secondary access point (AP) is used for multicast (common information) transmission to a number of secondary singleantenna receivers. The idea is that beamforming can be used to steer power towards the secondary receivers while limiting sidelobes that cause interference to primary receivers. Various optimal formulations of beamforming are proposed, motivated by different “cohabitation ” scenarios, including robust designs that are applicable with inaccurate or limited channel state information at the secondary AP. These formulations are NPhard computational problems; yet it is shown how convex approximationbased multicast beamforming tools (originally developed without regard to primary interference constraints) can be adapted to work in a spectrum underlay context. Extensive simulation results demonstrate the effectiveness of the proposed approaches and provide insights on the tradeoffs between different design criteria. Index Terms—Beamforming, multicasting, secondary spectrum usage, semidefinite programming (SDP). I.
Complex Quadratic Optimization and Semidefinite Programming
 SIAM Journal on Optimization
, 2006
"... In this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the Hermitian complex form. A special case of the problem that we study corresponds to the max3cut model used in a recent paper of Goemans and Williamson. We first develop a closedform f ..."
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Cited by 31 (12 self)
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In this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the Hermitian complex form. A special case of the problem that we study corresponds to the max3cut model used in a recent paper of Goemans and Williamson. We first develop a closedform formula to compute the probability of a complexvalued normally distributed bivariate random vector to be in a given angular region. This formula allows us to compute the expected value of a randomized (with a specific rounding rule) solution based on the optimal solution of the complex SDP relaxation problem. In particular, we study the limit of that model, in which the problem remains NPhard. We show that if the objective is to maximize a positive semidefinite Hermitian form, then the randomizationrounding procedure guarantees a worstcase performance ratio of π/4 ≈ 0.7854, which is better than the ratio of 2/π ≈ 0.6366 for its counterpart in the real case due to Nesterov. Furthermore, if the objective matrix is realvalued positive semidefinite with nonpositive offdiagonal elements, then the performance ratio improves to 0.9349.
Optimal power flow over tree networks
 PROCEEDINGS OF THE FORTHNINTH ANNUAL ALLERTON CONFERENCE
, 2011
"... The optimal power flow (OPF) problem is critical to power system operation but it is generally nonconvex and therefore hard to solve. Recently, a sufficient condition has been found under which OPF has zero duality gap, which means that its solution can be computed efficiently by solving the conve ..."
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Cited by 28 (12 self)
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The optimal power flow (OPF) problem is critical to power system operation but it is generally nonconvex and therefore hard to solve. Recently, a sufficient condition has been found under which OPF has zero duality gap, which means that its solution can be computed efficiently by solving the convex dual problem. In this paper we simplify this sufficient condition through a reformulation of the problem and prove that the condition is always satisfied for a tree network provided we allow oversatisfaction of load. The proof, cast as a complex semidefinite program, makes use of the fact that if the underlying graph of an n n Hermitian positive semidefinite matrix is a tree, then the matrix has rank at least n  1.
Quadratically constrained quadratic programs on acyclic graphs with application to power flow
, 2013
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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.
Convex Relaxation of Optimal Power Flow  Part I: Formulations and Equivalence
, 2014
"... This tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow models, formulates OPF and their relaxations in each model, and proves equivalence relations among them. ..."
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Cited by 13 (0 self)
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This tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow models, formulates OPF and their relaxations in each model, and proves equivalence relations among them. Part II presents sufficient conditions under which the convex relaxations are exact.
Principles of minimum variance robust adaptive beamforming design
 Elsevier Signal Processing
, 2012
"... a b s t r a c t Robustness is typically understood as an ability of adaptive beamforming algorithm to achieve high performance in the situations with imperfect, incomplete, or erroneous knowledge about the source, propagation media, and antenna array. It is also desired to achieve high performance ..."
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Cited by 8 (0 self)
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a b s t r a c t Robustness is typically understood as an ability of adaptive beamforming algorithm to achieve high performance in the situations with imperfect, incomplete, or erroneous knowledge about the source, propagation media, and antenna array. It is also desired to achieve high performance with as little as possible prior information. In the last decade, several fruitful principles to minimum variance distortionless response (MVDR) robust adaptive beamforming (RAB) design have been developed and successfully applied to solve a number of problems in a wide range of applications. Such principles of MVDR RAB design are summarized here in a single paper. Prof. Gershman has actively participated in the development and applications of a number of such MVDR RAB design principles.
General Constrained Polynomial Optimization: an Approximation Approach
, 2009
"... In this paper, we consider approximation algorithms for optimizing a generic multivariate (inhomogeneous) polynomial function, subject to some fairly general constraints. The focus is on the design and analysis of polynomialtime approximation algorithms. First, we study the problem of maximizing a ..."
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Cited by 7 (4 self)
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In this paper, we consider approximation algorithms for optimizing a generic multivariate (inhomogeneous) polynomial function, subject to some fairly general constraints. The focus is on the design and analysis of polynomialtime approximation algorithms. First, we study the problem of maximizing a polynomial function over the Euclidean ball. A polynomialtime approximation algorithm is proposed for this problem with an assured (relative) worstcase performance ratio, which depends only on the dimensions of the model. The method is extended to optimize a polynomial function over the intersection of a finite number of cocentral ellipsoids. Likewise, an approximation algorithm is proposed with an assured (relative) worstcase performance ratio. Furthermore, the constraint set is relaxed to a general convex compact set. In particular, we propose an approximation algorithm with a (relative) worstcase performance ratio for polynomial optimization over some general set: for instance, a polytope. Finally, numerical results are reported, revealing remarkably good practical performance of the proposed algorithms for solving some randomly generated test instances.
Design of Optimized Radar Codes with a Peak to Average Power Ratio Constraint
 SUBMITTED TO IEEE TRANS. ON SIGNAL PROCESSING
, 2010
"... This paper considers the problem of radar waveform design in the presence of colored Gaussian disturbance under a Peak to Average power Ratio (PAR) and an energy constraint. First of all, we focus on the selection of the radar signal optimizing the Signal to Noise Ratio (SNR) in correspondence of a ..."
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Cited by 5 (1 self)
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This paper considers the problem of radar waveform design in the presence of colored Gaussian disturbance under a Peak to Average power Ratio (PAR) and an energy constraint. First of all, we focus on the selection of the radar signal optimizing the Signal to Noise Ratio (SNR) in correspondence of a given expected target Doppler frequency (Algorithm 1). Then, through a maxmin approach, we make robust the technique with respect to the received Doppler (Algorithm 2), namely we optimize the worst case SNR under the same constraints as in the previous problem. Since Algorithms 1 and 2 do not impose any condition on the waveform phase, we also devise their phase quantized versions (Algorithms 3 and 4 respectively), which force the waveform phase to lie within a finite alphabet. All the problems are formulated in terms of nonconvex quadratic optimization programs with either a finite or an infinite number of quadratic constraints. We prove that these problems are NPhard and, hence, introduce design techniques, relying on Semidefinite Programming (SDP) relaxation and randomization as well as on the theory of trigonometric polynomials, providing high quality suboptimal solutions with a polynomial time computational complexity. Finally, we analyze the performance of the new waveform design algorithms