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65
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 161 (11 self)
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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, 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
Optimal Beamforming for TwoWay MultiAntenna Relay Channel with Analogue Network Coding
, 2009
"... This paper studies the wireless twoway relay channel (TWRC), where two source nodes, S1 and S2, exchange information through an assisting relay node, R. It is assumed that R receives the sum signal from S1 and S2 in one timeslot, and then amplifies and forwards the received signal to both S1 and S ..."
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Cited by 90 (6 self)
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This paper studies the wireless twoway relay channel (TWRC), where two source nodes, S1 and S2, exchange information through an assisting relay node, R. It is assumed that R receives the sum signal from S1 and S2 in one timeslot, and then amplifies and forwards the received signal to both S1 and S2 in the next timeslot. By applying the principle of analogue network coding (ANC), each of S1 and S2 cancels the socalled “selfinterference ” in the received signal from R and then decodes the desired message. Assuming that S1 and S2 are each equipped with a single antenna and R with multiantennas, this paper analyzes the capacity region of the ANCbased TWRC with linear processing (beamforming) at R. The capacity region contains all the achievable bidirectional ratepairs of S1 and S2 under the given transmit power constraints at S1, S2, and R. We present the optimal relay beamforming structure as well as an efficient algorithm to compute the optimal beamforming matrix based on convex optimization techniques. Lowcomplexity suboptimal relay beamforming schemes are also presented, and their achievable rates are compared against the capacity with the optimal scheme.
A survey of the Slemma
 SIAM Review
"... Abstract. In this survey we review the many faces of the Slemma, a result about the correctness of the Sprocedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as ..."
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Cited by 63 (1 self)
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Abstract. In this survey we review the many faces of the Slemma, a result about the correctness of the Sprocedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as well. These were all active research areas, but as there was little interaction between researchers in these different areas, their results remained mainly isolated. Here we give a unified analysis of the theory by providing three different proofs for the Slemma and revealing hidden connections with various areas of mathematics. We prove some new duality results and present applications from control theory, error estimation, and computational geometry. Key words. Slemma, Sprocedure, control theory, nonconvex theorem of alternatives, numerical range, relaxation theory, semidefinite optimization, generalized convexities
Complex matrix decomposition and quadratic programming
, 2006
"... This paper studies the possibilities of the Linear Matrix Inequality (LMI) characterization of the matrix cones formed by nonnegative complex Hermitian quadratic functions over specific domains in the complex space. In its real case analog, such studies were conducted in Sturm and Zhang [11]. In thi ..."
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Cited by 59 (16 self)
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This paper studies the possibilities of the Linear Matrix Inequality (LMI) characterization of the matrix cones formed by nonnegative complex Hermitian quadratic functions over specific domains in the complex space. In its real case analog, such studies were conducted in Sturm and Zhang [11]. In this paper it is shown that stronger results can be obtained for the complex Hermitian case. In particular, we show that the matrix rankone decomposition result of Sturm and Zhang [11] can be strengthened for the complex Hermitian matrices. As a consequence, it is possible to characterize several new matrix copositive cones (over specific domains) by means of LMI. As examples of the potential application of the new rankone decomposition result, we present an upper bound on the lowest rank among all the optimal solutions for a standard complex SDP problem, and offer alternative proofs for a result of Hausdorff [5] and a result of Brickman [3] on the joint numerical range.
Exact and approximate solution of source localization problems
 IEEE Trans. Signal Processing
, 2007
"... Abstract—We consider least squares (LS) approaches for locating a radiating source from range measurements (which we call RLS) or from rangedifference measurements (RDLS) collected using an array of passive sensors. We also consider LS approaches based on squared range observations (SRLS) and ba ..."
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Cited by 46 (1 self)
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Abstract—We consider least squares (LS) approaches for locating a radiating source from range measurements (which we call RLS) or from rangedifference measurements (RDLS) collected using an array of passive sensors. We also consider LS approaches based on squared range observations (SRLS) and based on squared rangedifference measurements (SRDLS). Despite the fact that the resulting optimization problems are nonconvex, we provide exact solution procedures for efficiently computing the SRLS and SRDLS estimates. Numerical simulations suggest that the exact SRLS and SRDLS estimates outperform existing approximations of the SRLS and SRDLS solutions as well as approximations of the RLS and RDLS solutions which are based on a semidefinite relaxation. Index Terms—Efficiently and globally optimal solution, generalized trust region subproblems (GTRS), least squares, nonconvex, quadratic function minimization, range measurements, rangedifference measurements, single quadratic constraint, source localization, squared range observations. I.
Strong Duality in Nonconvex Quadratic Optimization with Two Quadratic Constraints
 SIAM JOURNAL ON OPTIMIZATION
, 2006
"... We consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints. When the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions. We then develop a connection betwe ..."
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Cited by 44 (9 self)
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We consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints. When the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions. We then develop a connection between the image of the real and complex spaces under a quadratic mapping, which together with the results in the complex case lead to a condition that ensures strong duality in the real setting. Preliminary numerical simulations suggest that for random instances of the extended trust region subproblem, the sufficient condition is satisfied with a high probability. Furthermore, we show that the sufficient condition is always satisfied in two classes of nonconvex quadratic problems. Finally, we discuss an application of our results to robust least squares problems.
BiQuadratic Optimization over Unit Spheres and Semidefinite Programming Relaxations
, 2008
"... Abstract. This paper studies the socalled biquadratic optimization over unit spheres min x∈R n,y∈R m bijklxiyjxkyl ..."
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Cited by 32 (15 self)
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Abstract. This paper studies the socalled biquadratic optimization over unit spheres min x∈R n,y∈R m bijklxiyjxkyl
Convex sets with semidefinite representation. Optimization Online
, 2006
"... Abstract. We provide a sufficient condition on a class of compact basic semialgebraic sets K ⊂ R n for their convex hull co(K) to have a semidefinite representation (SDr). This SDr is explicitly expressed in terms of the polynomials gj that define K. Examples are provided. We also provide an approxi ..."
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Cited by 29 (1 self)
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Abstract. We provide a sufficient condition on a class of compact basic semialgebraic sets K ⊂ R n for their convex hull co(K) to have a semidefinite representation (SDr). This SDr is explicitly expressed in terms of the polynomials gj that define K. Examples are provided. We also provide an approximate SDr; that is, for every fixed ɛ> 0, there is a convex set Kɛ such that co(K) ⊆ Kɛ ⊆ co(K) + ɛB (where B is the unit ball of R n), and Kɛ has an explicit SDr in terms of the gj’s. For convex and compact basic semialgebraic sets K defined by concave polynomials, we provide a simpler explicit SDr when the nonnegative Lagrangian Lf associated with K and any linear f ∈ R[X] is a sum of squares. We also provide an approximate SDr specific to the convex case. 1.
Quadratic matrix programming
 SIAM J. Optim
"... We introduce and study a special class of nonconvex quadratic problems in which the objective and constraint functions have the form f(X) = Tr(X T AX) + 2Tr(B T X) + c, X ∈ R n×r The latter formulation is termed quadratic matrix programming (QMP) of order r. We construct a specially devised semidef ..."
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Cited by 27 (2 self)
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We introduce and study a special class of nonconvex quadratic problems in which the objective and constraint functions have the form f(X) = Tr(X T AX) + 2Tr(B T X) + c, X ∈ R n×r The latter formulation is termed quadratic matrix programming (QMP) of order r. We construct a specially devised semidefinite relaxation (SDR) and dual for the QMP problem and show that under some mild conditions strong duality holds for QMP problems with at most r constraints. Using a result on the equivalence of two characterizations of the nonnegativity property of quadratic functions of the above form, we are able to compare the constructed SDR and dual problems to other known SDR and dual formulations of the problem. An application to robust least squares problems is discussed. 1