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109
Theory and applications of Robust Optimization
, 2007
"... In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most pr ..."
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Cited by 110 (16 self)
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In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most prominent theoretical results of RO over the past decade, we will also present some recent results linking RO to adaptable models for multistage decisionmaking problems. Finally, we will highlight successful applications of RO across a wide spectrum of domains, including, but not limited to, finance, statistics, learning, and engineering.
A robust maximin approach for MIMO communications with imperfect channel state information based on convex optimization
 IEEE Trans. Signal Processing
, 2006
"... Abstract—This paper considers a wireless communication system with multiple transmit and receive antennas, i.e., a multipleinputmultipleoutput (MIMO) channel. The objective is to design the transmitter according to an imperfect channel estimate, where the errors are explicitly taken into account ..."
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Cited by 50 (5 self)
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Abstract—This paper considers a wireless communication system with multiple transmit and receive antennas, i.e., a multipleinputmultipleoutput (MIMO) channel. The objective is to design the transmitter according to an imperfect channel estimate, where the errors are explicitly taken into account to obtain a robust design under the maximin or worst case philosophy. The robust transmission scheme is composed of an orthogonal space–time block code (OSTBC), whose outputs are transmitted through the eigenmodes of the channel estimate with an appropriate power allocation among them. At the receiver, the signal is detected assuming a perfect channel knowledge. The optimization problem corresponding to the design of the power allocation among the estimated eigenmodes, whose goal is the maximization of the signaltonoise ratio (SNR), is transformed to a simple convex problem that can be easily solved. Different sources of errors are considered in the channel estimate, such as the Gaussian noise from the estimation process and the errors from the quantization of the channel estimate, among others. For the case of Gaussian noise, the robust power allocation admits a closedform expression. Finally, the benefits of the proposed design are evaluated and compared with the pure OSTBC and nonrobust approaches. Index Terms—Antenna arrays, beamforming, convex optimization theory, maximum optimization problems, multipleinput multipleoutput (MIMO) systems, saddle point, space–time coding, worstcase robust designs. I.
MIMO Radar Space–Time Adaptive Processing Using Prolate Spheroidal Wave Functions
, 2008
"... In the traditional transmitting beamforming radar system, the transmitting antennas send coherent waveforms which form a highly focused beam. In the multipleinput multipleoutput (MIMO) radar system, the transmitter sends noncoherent (possibly orthogonal) broad (possibly omnidirectional) waveforms ..."
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Cited by 37 (8 self)
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In the traditional transmitting beamforming radar system, the transmitting antennas send coherent waveforms which form a highly focused beam. In the multipleinput multipleoutput (MIMO) radar system, the transmitter sends noncoherent (possibly orthogonal) broad (possibly omnidirectional) waveforms. These waveforms can be extracted at the receiver by a matched filterbank. The extracted signals can be used to obtain more diversity or to improve the spatial resolution for clutter. This paper focuses on space–time adaptive processing (STAP) for MIMO radar systems which improves the spatial resolution for clutter. With a slight modification, STAP methods developed originally for the singleinput multipleoutput (SIMO) radar (conventional radar) can also be used in MIMO radar. However, in the MIMO radar, the rank of the jammerandclutter subspace becomes very large, especially the jammer subspace. It affects both the complexity and the convergence of the STAP algorithm. In this paper, the clutter space and its rank in the MIMO radar are explored. By using the geometry of the problem rather than data, the clutter subspace can be represented using prolate spheroidal wave functions (PSWF). A new STAP algorithm is also proposed. It computes the clutter space using the PSWF and utilizes the blockdiagonal property of the jammer covariance matrix. Because of fully utilizing the geometry and the structure of the covariance matrix, the method has very good SINR performance and low computational complexity.
Convex conic formulations of robust downlink precoder designs with quality of service constraints
 IEEE J. Select. Topics Signal Processing
, 2007
"... We consider the design of linear precoders (beamformers) for broadcast channels with Quality of Service (QoS) constraints for each user, in scenarios with uncertain channel state information (CSI) at the transmitter. We consider a deterministicallybounded model for the channel uncertainty of each u ..."
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Cited by 35 (2 self)
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We consider the design of linear precoders (beamformers) for broadcast channels with Quality of Service (QoS) constraints for each user, in scenarios with uncertain channel state information (CSI) at the transmitter. We consider a deterministicallybounded model for the channel uncertainty of each user, and our goal is to design a robust precoder that minimizes the total transmission power required to satisfy the users ’ QoS constraints for all channels within a specified uncertainty region around the transmitter’s estimate of each user’s channel. Since this problem is not known to be computationally tractable, we will derive three conservative design approaches that yield convex and computationallyefficient restrictions of the original design problem. The three approaches yield semidefinite program (SDP) formulations that offer different tradeoffs between the degree of conservatism and the size of the SDP. We will also show how these conservative approaches can be used to derive efficientlysolvable quasiconvex restrictions of some related design problems, including the robust counterpart to the problem of maximizing the minimum signaltointerferenceplusnoiseratio (SINR) subject to a given power constraint. Our simulation results indicate that in the presence of uncertain CSI the proposed approaches can satisfy the users ’ QoS requirements for a significantly larger set of uncertainties than existing methods, and require less transmission power to do so.
Radarbased breast cancer detection using a hemispherical antenna array—experimental results
 IEEE Transactions on Antennas and Propagation
, 2009
"... This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: ..."
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Cited by 29 (2 self)
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This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available:
Robust power allocation designs for multiuser and multiantenna downlink communication systems through convex optimization
 IEEE J. Sel. Areas Commun
, 2007
"... Abstract — In this paper, we study the design of the transmitter in the downlink of a multiuser and multiantenna wireless communications system, considering the realistic scenario where only an imperfect estimate of the actual channel is available at both communication ends. Precisely, the actual ch ..."
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Cited by 24 (1 self)
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Abstract — In this paper, we study the design of the transmitter in the downlink of a multiuser and multiantenna wireless communications system, considering the realistic scenario where only an imperfect estimate of the actual channel is available at both communication ends. Precisely, the actual channel is assumed to be inside an uncertainty region around the channel estimate, which models the imperfections of the channel knowledge that may arise from, e.g., estimation Gaussian errors, quantization effects, or combinations of both sources of errors. In this context, our objective is to design a robust power allocation among the information symbols that are to be sent to the users such that the total transmitted power is minimized, while maintaining the necessary quality of service to obtain reliable communication links between the base station and the users for any possible realization of the actual channel inside the uncertainty region. This robust power allocation is obtained as the solution to a convex optimization problem, which, in general, can be numerically solved in a very efficient way, and even for a particular case of the uncertainty region, a quasiclosed form solution can be found. Finally, the goodness of the robust proposed transmission scheme is presented through numerical results. Index Terms — Robust designs, imperfect CSI, multiantenna systems, broadcast channel, convex optimization.
Convex Optimizationbased Beamforming  From receive to transmit and network designs
, 2010
"... In this article, an overview of advanced convex optimization approaches to multisensor beamforming is presented, and connections are drawn between different types of optimizationbased beamformers that apply to a broad class of receive, transmit, and network beamformer design problems. It is demons ..."
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Cited by 21 (0 self)
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In this article, an overview of advanced convex optimization approaches to multisensor beamforming is presented, and connections are drawn between different types of optimizationbased beamformers that apply to a broad class of receive, transmit, and network beamformer design problems. It is demonstrated that convex optimization provides an indispensable set of tools for beamforming, enabling rigorous formulation and effective solution of both longstanding and emerging design problems.
On the relationship between robust minimum variance beamformers with probabilistic and worstcase distrortionless response constraints
 IEEE Trans. Signal Process
, 2008
"... Abstract—An interesting relationship between the probabilityconstrained and worstcase optimization based robust minimum variance (MV) beamformers has been discovered. It is shown that both in the cases of circularly symmetric Gaussian and worstcase distributions of the steering vector mismatch, ..."
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Cited by 18 (7 self)
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Abstract—An interesting relationship between the probabilityconstrained and worstcase optimization based robust minimum variance (MV) beamformers has been discovered. It is shown that both in the cases of circularly symmetric Gaussian and worstcase distributions of the steering vector mismatch, the probabilityconstrained robust MV beamforming problem can be tightly approximated as a convex secondorder cone programming (SOCP) problem. The latter problem is mathematically equivalent to that resulting from the deterministic worstcase approach and, therefore, probabilityconstrained beamformers can be interpreted and implemented using their deterministic worstcase counterparts. However, an important advantage of the developed probabilityconstrained MV beamformers with respect to their standard worstcase counterparts is that the former approaches enable to explicitly quantify the parameters of the uncertainty region in terms of the beamformer outage probability. Index Terms—Probabilistic distortionless response constraints, robust minimum variance beamforming. I.
Cuttingset methods for robust convex optimization with pessimizing oracles
 DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING, UNIVERSITY OF CALIFORNIA, SAN DIEGO. FROM
, 2011
"... We consider a general worstcase robust convex optimization problem, with arbitrary dependence on the uncertain parameters, which are assumed to lie in some given set of possible values. We describe a general method for solving such a problem, which alternates between optimization and worstcase ana ..."
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Cited by 17 (5 self)
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We consider a general worstcase robust convex optimization problem, with arbitrary dependence on the uncertain parameters, which are assumed to lie in some given set of possible values. We describe a general method for solving such a problem, which alternates between optimization and worstcase analysis. With exact worstcase analysis, the method is shown to converge to a robust optimal point. With approximate worstcase analysis, which is the best we can do in many practical cases, the method seems to work very well in practice, subject to the errors in our worstcase analysis. We give variations on the basic method that can give enhanced convergence, reduce data storage, or improve other algorithm properties. Numerical simulations suggest that the method finds a quite robust solution within a few tens of steps; using warmstart techniques in the optimization steps reduces the overall effort to a modest multiple of solving a nominal problem, ignoring the parameter variation. The method is illustrated with several application examples.
Quadratically Constrained Beamforming Robust Against DirectionofArrival Mismatch
 IEEE TRANS. ON SIGNAL PROCESSING
, 2007
"... It is well known that the performance of the minimum variance distortionless response (MVDR) beamformer is very sensitive to steering vector mismatch. Such mismatches can occur as a result of directionofarrival (DOA) errors, local scattering, nearfar spatial signature mismatch, waveform distorti ..."
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Cited by 15 (1 self)
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It is well known that the performance of the minimum variance distortionless response (MVDR) beamformer is very sensitive to steering vector mismatch. Such mismatches can occur as a result of directionofarrival (DOA) errors, local scattering, nearfar spatial signature mismatch, waveform distortion, source spreading, imperfectly calibrated arrays and distorted antenna shape. In this paper, an adaptive beamformer that is robust against the DOA mismatch is proposed. This method imposes two quadratic constraints such that the magnitude responses of two steering vectors exceed unity. Then, a diagonal loading method is used to force the magnitude responses at the arrival angles between these two steering vectors to exceed unity. Therefore, this method can always force the gains at a desired range of angles to exceed a constant level while suppressing the interferences and noise. A closedform solution to the proposed minimization problem is introduced, and the diagonal loading factor can be computed systematically by a proposed algorithm. Numerical examples show that this method has excellent signaltointerferenceplusnoise ratio performance and a complexity comparable to the standard MVDR beamformer.