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177
sdpsol: A Parser/Solver for Semidefinite Programs with Matrix Structure
 In Recent advances in LMI methods for control
, 1995
"... . A variety of analysis and design problems in control, communication and information theory, statistics, combinatorial optimization, computational geometry, circuit design, and other fields can be expressed as semidefinite programming problems (SDPs) or determinant maximization problems (maxdet pr ..."
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Cited by 42 (16 self)
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. A variety of analysis and design problems in control, communication and information theory, statistics, combinatorial optimization, computational geometry, circuit design, and other fields can be expressed as semidefinite programming problems (SDPs) or determinant maximization problems (maxdet problems). These problems often have matrix structure, i.e., some of the optimization variables are matrices. This matrix structure has two important practical ramifications: first, it makes the job of translating the problem into a standard SDP or maxdet format tedious, and, second, it opens the possibility of exploiting the structure to speed up the computation. In this paper we describe the design and implementation of sdpsol, a parser/solver for SDPs and maxdet problems. sdpsol allows problems with matrix structure to be described in a simple, natural, and convenient way. Although the current implementation of sdpsol does not exploit matrix structure in the solution algorithm, the languag...
Grasp analysis as linear matrix inequality problems
 IEEE Transactions on Robotics and Automation
, 2000
"... ..."
On the Capacity of Multiple Input Multiple Output Broadcast Channels
 In Proceedings of Int. Conf. Commun
, 2002
"... We consider a twouser multiple input multiple output (MIMO) Gaussian broadcast channel (BC), where the transmitter has t transmit antennas and receivers have r1 ; r2 antennas respectively. Since the MIMO broadcast channel is in general a nondegraded broadcast channel, its capacity region remains a ..."
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Cited by 33 (9 self)
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We consider a twouser multiple input multiple output (MIMO) Gaussian broadcast channel (BC), where the transmitter has t transmit antennas and receivers have r1 ; r2 antennas respectively. Since the MIMO broadcast channel is in general a nondegraded broadcast channel, its capacity region remains an unsolved problem. In this paper, we establish a duality between what is termed the \dirty paper" region (or the CostaCaireShamaiYu achievable region) [5, 7] for the MIMO broadcast channel and the capacity region of the the MIMO multipleaccess channel (MAC), which is easy to compute. Using this duality, we greatly reduce the computation complexity required for obtaining the dirty paper achievable region for the MIMO BC. The duality also enables us to translate previously known results for the MIMO MAC (like iterative waterlling [7]) to the MIMO BC. We show that the dirty paper achievable region achieves the sumrate capacity of the MIMO BC by establishing that the sumrate point in this region equals an upperbound on the sum rate of the MIMO BC. I.
On efficient representation and computation of reachable sets for hybrid systems
 In HSCCâ€™2003, LNCS 2289
, 2003
"... Abstract. Computing reachable sets is an essential step in most analysis and synthesis techniques for hybrid systems. The representation of these sets has a deciding impact on the computational complexity and thus the applicability of these techniques. This paper presents a new approach for approxim ..."
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Cited by 28 (6 self)
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Abstract. Computing reachable sets is an essential step in most analysis and synthesis techniques for hybrid systems. The representation of these sets has a deciding impact on the computational complexity and thus the applicability of these techniques. This paper presents a new approach for approximating reachable sets using oriented rectangular hulls (ORHs), the orientations of which are determined by singular value decompositions of sample covariance matrices for sets of reachable states. The orientations keep the overapproximation of the reachable sets small in most cases with a complexity of low polynomial order with respect to the dimension of the continuous state space. We show how the use of ORHs can improve the efficiency of reachable set computation significantly for hybrid systems with nonlinear continuous dynamics.
LogDeterminant Relaxation for Approximate Inference in Discrete Markov Random Fields
, 2006
"... Graphical models are well suited to capture the complex and nonGaussian statistical dependencies that arise in many realworld signals. A fundamental problem common to any signal processing application of a graphical model is that of computing approximate marginal probabilities over subsets of nod ..."
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Cited by 27 (3 self)
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Graphical models are well suited to capture the complex and nonGaussian statistical dependencies that arise in many realworld signals. A fundamental problem common to any signal processing application of a graphical model is that of computing approximate marginal probabilities over subsets of nodes. This paper proposes a novel method, applicable to discretevalued Markov random fields (MRFs) on arbitrary graphs, for approximately solving this marginalization problem. The foundation of our method is a reformulation of the marginalization problem as the solution of a lowdimensional convex optimization problem over the marginal polytope. Exactly solving this problem for general graphs is intractable; for binary Markov random fields, we describe how to relax it by using a Gaussian bound on the discrete entropy and a semidefinite outer bound on the marginal polytope. This combination leads to a logdeterminant maximization problem that can be solved efficiently by interior point methods, thereby providing approximations to the exact marginals. We show how a slightly weakened logdeterminant relaxation can be solved even more efficiently by a dual reformulation. When applied to denoising problems in a coupled mixtureofGaussian model defined on a binary MRF with cycles, we find that the performance of this logdeterminant relaxation is comparable or superior to the widely used sumproduct algorithm over a range of experimental conditions.
MinimumVolume Enclosing Ellipsoids and Core Sets
 JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
, 2005
"... We study the problem of computing a (1 + #)approximation to the minimum volume enclosing ellipsoid of a given point set , p . Based on a simple, initial volume approximation method, we propose a modification of Khachiyan's firstorder algorithm. Our analysis leads to a slightly imp ..."
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Cited by 26 (4 self)
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We study the problem of computing a (1 + #)approximation to the minimum volume enclosing ellipsoid of a given point set , p . Based on a simple, initial volume approximation method, we propose a modification of Khachiyan's firstorder algorithm. Our analysis leads to a slightly improved complexity bound of O(nd (0, 1). As a byproduct, our algorithm returns a core set with the property that the minimum volume enclosing ellipsoid of provides a good approximation to that of S.
Multiple Antenna Channels with Partial Channel State Information at the Transmitter
 IEEE Trans. Wireless Commun
, 2004
"... We investigate transmission strategies for flatfading multiple antenna channels with transmit and receive antennas, and with channel state information (CSI) partially known to the transmitter. We start with an assumption that the first eigenvectors of , where 0 min( ) and is the channel matrix in , ..."
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Cited by 25 (6 self)
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We investigate transmission strategies for flatfading multiple antenna channels with transmit and receive antennas, and with channel state information (CSI) partially known to the transmitter. We start with an assumption that the first eigenvectors of , where 0 min( ) and is the channel matrix in , are available at the transmitter as partial spatial information of the channel. A beamforming method is proposed in which a beamforming matrix is determined from the eigenvectors in some predefined way; as a result, the receiver also knows the beamforming matrix. With this beamforming scheme, we develop a new multiple antenna system concept that provides a mechanism to reduce the amount of channel feedback information. This paper focuses on deriving the channel capacity of the multiple antenna channels employing the proposed beamforming and feedback methods. An important task for achieving capacity is the solution of interesting optimization problems for the optimal power allocation over the transmit symbols. The results show that the proposed methods lead to systems wherein the amount of feedback information can be significantly reduced with a minor sacrifice of achievable transmission rate.
Computation of Minimum Volume Covering Ellipsoids
 Operations Research
, 2003
"... We present a practical algorithm for computing the minimum volume ndimensional ellipsoid that must contain m given points a 1 , . . . , am . This convex constrained problem arises in a variety of applied computational settings, particularly in data mining and robust statistics. Its structur ..."
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Cited by 23 (0 self)
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We present a practical algorithm for computing the minimum volume ndimensional ellipsoid that must contain m given points a 1 , . . . , am . This convex constrained problem arises in a variety of applied computational settings, particularly in data mining and robust statistics. Its structure makes it particularly amenable to solution by interiorpoint methods, and it has been the subject of much theoretical complexity analysis. Here we focus on computation. We present a combined interiorpoint and activeset method for solving this problem. Our computational results demonstrate that our method solves very large problem instances (m = 30, 000 and n = 30) to a high degree of accuracy in under 30 seconds on a personal computer.
Quadratic Stabilization and Control of PiecewiseLinear Systems
 In Proc. American Control Conf
, 1998
"... We consider analysis and controller synthesis of piecewiselinear systems. The method is based on constructing quadratic and piecewisequadratic Lyapunov functions that prove stability and performance for the system. It is shown that proving stability and performance, or designing (statefeedback) c ..."
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Cited by 22 (3 self)
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We consider analysis and controller synthesis of piecewiselinear systems. The method is based on constructing quadratic and piecewisequadratic Lyapunov functions that prove stability and performance for the system. It is shown that proving stability and performance, or designing (statefeedback) controllers, can be cast as convex optimization problems involving linear matrix inequalities that can be solved very e ciently. A couple of simple examples are included to demonstrate applications of the methods described. Key words: Piecewiselinear systems, quadratic stabilization, linear matrix inequality (LMI). 1