A simple method for the design of an optimal two-channel orthogonal cyclic filterbank using semidefinite programming is presented. The criterion for optimality is to maximize the passband energy, or equivalently, to m...
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A simple method for the design of an optimal two-channel orthogonal cyclic filterbank using semidefinite programming is presented. The criterion for optimality is to maximize the passband energy, or equivalently, to minimize the stopband energy of the filter's impulse response. The objective function and orthogonality constraints are represented in terms of the cyclic autocorrelation sequence of the filter's impulse response. The convex formulation of the filter design problem is obtained by imposing the positive semidefinite property on the cyclic autocorrelation matrix. The solution of the semidefinite programming problem gives a class of optimal cyclic filters with unique discrete Fourier transform magnitude response. The design method is illustrated with an example.
Positive semidefinite Hankel matrices arise in many important applications. Some of their properties may be lost due to rounding or truncation errors incurred during evaluation. The problem is to find the nearest matr...
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Positive semidefinite Hankel matrices arise in many important applications. Some of their properties may be lost due to rounding or truncation errors incurred during evaluation. The problem is to find the nearest matrix to a given matrix to retrieve these properties. The problem is converted into a semidefinite programming problem as well as a problem comprising a semidefined program and second-order cone problem. The duality and optimality conditions are obtained and the primal-dual algorithm is outlined. Explicit expressions for a diagonal preconditioned and crossover criteria have been presented. Computational results are presented. A possibility for further improvement is indicated. (c) 2006 Elsevier B.V. All rights reserved.
A critical obstacle for ultra-wideband (UWB) communications is conformity to restrictions set on the allowed interference to other wireless devices. To this end, UWB signals have to comply with stringent constraints o...
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A critical obstacle for ultra-wideband (UWB) communications is conformity to restrictions set on the allowed interference to other wireless devices. To this end, UWB signals have to comply with stringent constraints on their emitted power, defined by the Federal Communications Commission spectral mask. Different UWB pulseshaper designs have been studied to meet the spectral mask, out of which an approach based on digital finite impulse response filter design via semidefinite programming has stood out. However, so far this approach has assumed an ideal basic analog pulse to use piece-wise constant constraints for the digital filter design. Since any practical analog pulse does not have a flat spectrum, using piece-wise constant constraints leads to considerable power loss. Avoiding such a loss has motivated us to implement the exact constraints through nonconstant piece-wise continuous bounds. Relative to the design assuming an ideal basic analog pulse, our design examples show that the transmission power can be enhanced considerably while obeying the spectral mask. Such an improvement comes with no extra cost of implementation complexity.
In this paper we define semidefinite packing programs and describe an algorithm to approximately solve these problems. semidefinite packing programs arise in many applications such as semidefinite programming relaxati...
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In this paper we define semidefinite packing programs and describe an algorithm to approximately solve these problems. semidefinite packing programs arise in many applications such as semidefinite programming relaxations for combinatorial optimization problems, sparse principal component analysis, and sparse variance unfolding techniques for dimension reduction. Our algorithm exploits the structural similarity between semidefinite packing programs and linear packing programs.
The supervision of local specific absorption rate (SAR) in parallel transmission applications in MRI is crucial. One existing approach is to use electromagnetic simulations including human anatomical models and to pre...
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The supervision of local specific absorption rate (SAR) in parallel transmission applications in MRI is crucial. One existing approach is to use electromagnetic simulations including human anatomical models and to precalculate the electric field distributions of each individual channel. These can be superposed later with respect to certain combined excitations under investigation, and the local SAR distribution can be evaluated. Local SAR maxima can be obtained by exhaustive search over all investigated subvolumes of the body model. Practical challenges arise for the adequate handling and comparing of precalculated field distributions as long as the expected combined radiofrequency excitations are still undetermined. Worst-case approximations for local SAR lead to significant radiofrequency pulse performance limitations. Optimizing local SAR in radiofrequency pulse design using constraints for each subvolume is impractical. A method is proposed to significantly reduce the complexity without restriction to particular radiofrequency excitations. By constructing several matrices, it becomes sufficient to consider only these so-called Virtual Observation Points for an adequate, conservative estimation of the maximum local SAR. The applied techniques involve concepts of vector optimization as well as semidefinite programming. Magn Reson Med 66:1468-1476, 2011. (C) 2011 Wiley Periodicals, Inc.
Linear stochastic programming provides a flexible toolbox for analyzing real-life decision situations, but it can become computationally cumbersome when recourse decisions are involved. The latter are usually modeled ...
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Linear stochastic programming provides a flexible toolbox for analyzing real-life decision situations, but it can become computationally cumbersome when recourse decisions are involved. The latter are usually modeled as decision rules, i.e., functions of the uncertain problem data. It has recently been argued that stochastic programs can quite generally be made tractable by restricting the space of decision rules to those that exhibit a linear data dependence. In this paper, we propose an efficient method to estimate the approximation error introduced by this rather drastic means of complexity reduction: we apply the linear decision rule restriction not only to the primal but also to a dual version of the stochastic program. By employing techniques that are commonly used in modern robust optimization, we show that both arising approximate problems are equivalent to tractable linear or semidefinite programs of moderate sizes. The gap between their optimal values estimates the loss of optimality incurred by the linear decision rule approximation. Our method remains applicable if the stochastic program has random recourse and multiple decision stages. It also extends to cases involving ambiguous probability distributions.
An algorithm is given for the problem of finding the finest simultaneous block-diagonalization of a given set of square matrices. This problem has been studied independently in the area of semidefinite programming and...
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An algorithm is given for the problem of finding the finest simultaneous block-diagonalization of a given set of square matrices. This problem has been studied independently in the area of semidefinite programming and independent component analysis. The proposed algorithm considers the commutant algebra of the matrix *-algebra generated by the given matrices. It is simpler than other existing methods, and has the capability of controlling numerical errors. Some numerical examples are presented to demonstrate its merits.
Given a covariance matrix, we consider the problem of maximizing the variance explained by a particular linear combination of the input variables while constraining the number of nonzero coefficients in this combinati...
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Given a covariance matrix, we consider the problem of maximizing the variance explained by a particular linear combination of the input variables while constraining the number of nonzero coefficients in this combination. This problem arises in the decomposition of a covariance matrix into sparse factors or sparse principal component analysis (PCA), and has wide applications ranging from biology to finance. We use a modification of the classical variational representation of the largest eigenvalue of a symmetric matrix, where cardinality is constrained, and derive a semidefinite programming-based relaxation for our problem. We also discuss Nesterov's smooth minimization technique applied to the semidefinite program arising in the semidefinite relaxation of the sparse PCA problem. The method has complexity O(n(4) root log(n)/epsilon), where n is the size of the underlying covariance matrix and e is the desired absolute accuracy on the optimal value of the problem.
We recommend an implementation of the Markowitz problem to generate stable portfolios with respect to perturbations of the problem parameters. The stability is obtained proposing novel calibrations of the covariance m...
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We recommend an implementation of the Markowitz problem to generate stable portfolios with respect to perturbations of the problem parameters. The stability is obtained proposing novel calibrations of the covariance matrix between the returns that can be cast as convex or quasiconvex optimization problems. A statistical study as well as a sensitivity analysis of the Markowitz problem allow us to justify these calibrations. Our approach can be used to do a global and explicit sensitivity analysis of a class of quadratic optimization problems. Numerical simulations finally show the benefits of the proposed calibrations using real data.
Cooperative transmission in relay networks is considered, in which a source transmits to its destination with the help of a set of cooperating nodes. The source first transmits locally. The cooperating nodes that rece...
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Cooperative transmission in relay networks is considered, in which a source transmits to its destination with the help of a set of cooperating nodes. The source first transmits locally. The cooperating nodes that receive the source signal retransmit a weighted version of it in an amplify-and-forward (AF) fashion. Assuming knowledge of the second-order statistics of the channel state information, beamforming weights are determined so that the signal-to-noise ratio (SNR) at the destination is maximized subject to two different power constraints, i.e., a total (source and relay) power constraint, and individual relay power constraints. For the former constraint, the original problem is transformed into a problem of one variable, which can be solved via Newton's method. For the latter constraint, this problem is solved completely. It is shown that the semidefinite programming (SDP) relaxation of the original problem always has a rank one solution, and hence the original problem is equivalent to finding the rank one solution of the SDP problem. An explicit construction of such a rank one solution is also provided. Numerical results are presented to illustrate the proposed theoretical findings.
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