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39
Disciplined convex programming
 Global Optimization: From Theory to Implementation, Nonconvex Optimization and Its Application Series
, 2006
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Semidefinite Programming Relaxations and Algebraic Optimization in Control
 EUROPEAN JOURNAL OF CONTROL (2003)9:307321
, 2003
"... We present an overview of the essential elements of semidefinite programming as a computational tool for the analysis of systems and control problems. We make particular emphasis on general duality properties as providing suboptimality or infeasibility certificates. Our focus is on the exciting deve ..."
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Cited by 31 (5 self)
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We present an overview of the essential elements of semidefinite programming as a computational tool for the analysis of systems and control problems. We make particular emphasis on general duality properties as providing suboptimality or infeasibility certificates. Our focus is on the exciting developments which have occured in the last few years, including robust optimization, combinatorial optimization, and algebraic methods such as sumofsquares. These developments are illustrated with examples of applications to control systems.
Semidefinite Programming Duality and Linear TimeInvariant Systems
, 2003
"... Several important problems in control theory can be reformulated as semidefinite programming problems, i.e., minimization of a linear objective subject to Linear Matrix Inequality (LMI) constraints. From convex optimization duality theory, conditions for infeasibility of the LMIs as well as dual opt ..."
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Cited by 28 (2 self)
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Several important problems in control theory can be reformulated as semidefinite programming problems, i.e., minimization of a linear objective subject to Linear Matrix Inequality (LMI) constraints. From convex optimization duality theory, conditions for infeasibility of the LMIs as well as dual optimization problems can be formulated. These can in turn be reinterpreted in control or system theoretic terms, often yielding new results or new proofs for existing results from control theory. We explore such connections for a few problems associated with linear timeinvariant systems. 1
Optimal waveform design for UWB radios
 George Mason University
, 2004
"... Abstract—With transmit power spectra strictly limited by regulatory spectral masks, the emerging ultrawideband (UWB) communication systems call for judicious pulse shape design in order to achieve optimal spectrum utilization, spectral mask compatibility, and coexistence with other wireless service ..."
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Cited by 22 (4 self)
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Abstract—With transmit power spectra strictly limited by regulatory spectral masks, the emerging ultrawideband (UWB) communication systems call for judicious pulse shape design in order to achieve optimal spectrum utilization, spectral mask compatibility, and coexistence with other wireless services. Meanwhile, orthogonal pulse sets are often desired in order to apply highrate multidimensional modulation and (carrierfree) orthogonal frequencydivision multiple access. Motivated by these considerations, we suggest a digital finite impulse response (FIR) filter approach to synthesizing UWB pulses and propose filter design techniques by which optimal waveforms that satisfy the spectral mask can be efficiently obtained. For single pulse design, we develop a convex formulation for the design of the FIR filter coefficients that maximize the spectrum utilization efficiency in terms of both the bandwidth and power allowed by the spectral mask. For orthogonal pulse design, a sequential strategy is derived to formulate the overall pulse design problem as a set of convex subproblems, which are then solved in a sequential manner to yield a set of mutually orthogonal pulses. Our design techniques not only provide waveforms with high spectrum utilization and guaranteed spectral mask compliance but also permit simple modifications that can accommodate several other system objectives. Index Terms—Digital pulse design, finite impulse response (FIR) filter, ultrawideband communications. I.
On Nesterov’s approach to semiinfinite programming
 Acta Applicandae Mathematicae
, 2002
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Interiorpoint algorithms for sumofsquares optimization of multidimensional trigonometric polynomials
 In Proceedings of the International Conference on Acoustics, Speech, and Signal Processing
, 2007
"... A wide variety of optimization problems involving nonnegative polynomials or trigonometric polynomials can be formulated as convex optimization problems by expressing (or relaxing) the constraints using sumofsquares representations. The semidefinite programming problems that result from this formu ..."
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Cited by 5 (1 self)
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A wide variety of optimization problems involving nonnegative polynomials or trigonometric polynomials can be formulated as convex optimization problems by expressing (or relaxing) the constraints using sumofsquares representations. The semidefinite programming problems that result from this formulation are often difficult to solve due to the presence of large auxiliary matrix variables. In this paper we extend a recent technique for exploiting structure in semidefinite programs derived from sumofsquares expressions to multivariate trigonometric polynomials. The technique is based on an equivalent formulation using discrete Fourier transforms and leads to a very substantial reduction in the computational complexity. Numerical results are presented and a comparison is made with generalpurpose semidefinite programming algorithms. As an application, we consider a twodimensional FIR filter design problem. Index Terms — Optimization methods, Multidimensional digital filters, Discrete transforms
Efficient largescale filter/filterbank design via LMI characterization of trigonometric curves
 in Proc. of IEEE Inter. Conf. on Acoustics, Speech and Signal Processing (ICASSP 05
"... Abstract—Many filter and filterbank design problems can be posed as the optimization of linear or convex quadratic objectives over trigonometric semiinfinite constraints. Recent advances in design methodology are based on various linear matrix inequality (LMI) characterizations of the semiinfinite ..."
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Cited by 5 (4 self)
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Abstract—Many filter and filterbank design problems can be posed as the optimization of linear or convex quadratic objectives over trigonometric semiinfinite constraints. Recent advances in design methodology are based on various linear matrix inequality (LMI) characterizations of the semiinfinite constraints, and semidefinite programming (SDP) solutions. Despite these advances, the design of filters of several hundredth order, which typically arise in multicarrier communication and signal compression, cannot be accommodated. This hurdle is due mainly to the large number of additional variables incurred in the LMI characterizations. This paper proposes a novel LMI characterization of the semiinfinite constraints that involves additional variables of miminal dimensions. Consequently, the design of highorder filters required in practical applications can be achieved. Examples of designs of up to 1200tap filters are presented to verify the viability of the proposed approach. Index Terms—Filter and filter bank, semidefinite programming, trigonometric polynomial. a linear TSIC in the filter coefficients [19], [32] (more precisely, (2) leads to two linear TSICs [16]). More general constraints [1], [2], [37] involving bounds on the frequency response of the FIR filter are also expressible in terms of linear TSIC. Imposing a passband ripple of in the passband, and a stopband attenuation of in the stopband is equivalent to the following linear TSICs (2) I.
Multidimensional FIR filter design via trigonometric sumofsquares optimization
, 2007
"... We discuss a method for multidimensional FIR filter design via sumofsquares formulations of spectral mask constraints. The sumofsquares optimization problem is expressed as a semidefinite program with lowrank structure, by sampling the constraints using discrete cosine and sine transforms. The ..."
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Cited by 2 (0 self)
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We discuss a method for multidimensional FIR filter design via sumofsquares formulations of spectral mask constraints. The sumofsquares optimization problem is expressed as a semidefinite program with lowrank structure, by sampling the constraints using discrete cosine and sine transforms. The resulting semidefinite program is then solved by a customized primaldual interiorpoint method that exploits lowrank structure. This leads to a substantial reduction in the computational complexity, compared to generalpurpose semidefinite programming methods that exploit sparsity.
Gram pair parameterization of multivariate sumofsquares trigonometric polynomials
, 2006
"... In this paper we propose a parameterization of sumofsquares trigonometric polynomials with real coefficients, that uses two positive semidefinite matrices twice smaller than the unique matrix in the known Gram (trace) parameterization. Also, we formulate a Bounded Real Lemma for polynomials with ..."
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In this paper we propose a parameterization of sumofsquares trigonometric polynomials with real coefficients, that uses two positive semidefinite matrices twice smaller than the unique matrix in the known Gram (trace) parameterization. Also, we formulate a Bounded Real Lemma for polynomials with support in the positive orthant. We show that the new parameterization is clearly faster and thus can replace the old one in several design problems. 1.
Convex relaxation approach to the identification of the wienerhammerstein model
 in 47th IEEE Conference on Decision and Control
, 2008
"... Abstract — In this paper, an input/output system identification technique for the WienerHammerstein model and its feedback extension is proposed. In the proposed framework, the identification of the nonlinearity is nonparametric. The identification problem can be formulated as a nonconvex quadra ..."
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Abstract — In this paper, an input/output system identification technique for the WienerHammerstein model and its feedback extension is proposed. In the proposed framework, the identification of the nonlinearity is nonparametric. The identification problem can be formulated as a nonconvex quadratic program (QP). A convex semidefinite programming (SDP) relaxation is then formulated and solved to obtain a suboptimal solution to the original nonconvex QP. The convex relaxation turns out to be tight in most cases. Combined with the use of local search, high quality solutions to the WienerHammerstein identification can frequently be found. As an application example, randomly generated WienerHammerstein models are identified. 1 I.