We show how to exploit symmetries of a graph to efficiently compute the fastest mixing Markov chain on the graph (i.e., find the transition probabilities on the edges to minimize the second-largest eigenvalue modulus ...
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We show how to exploit symmetries of a graph to efficiently compute the fastest mixing Markov chain on the graph (i.e., find the transition probabilities on the edges to minimize the second-largest eigenvalue modulus of the transition probability matrix). Exploiting symmetry can lead to significant reduction in both the number of variables and the size of matrices in the corresponding semidefinite program, and thus enable numerical solution of large-scale instances that are otherwise computationally infeasible. We obtain analytic or semianalytic results for particular classes of graphs, such as edge-transitive and distance-transitive graphs. We describe two general approaches for symmetry exploitation, based on orbit theory and block-diagonalization, respectively, and establish a formal connection between them.
For a transfer function F(e(3 omega)) of order n, Kalman-Yakubovich-Popov (KYP) lemma characterizes a general intractable semi-infinite programming (SIP) condition by a tractable semidefinite programming (SDP) for the...
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For a transfer function F(e(3 omega)) of order n, Kalman-Yakubovich-Popov (KYP) lemma characterizes a general intractable semi-infinite programming (SIP) condition by a tractable semidefinite programming (SDP) for the entire frequency range. Some recent results generalize this lemma for a certain frequency interval. All these SDP characterizations are given at the expense of the introduced Lyapunov matrix variable of dimension n x n. Consequently, formulation and design of high dimensional problem is challenging. Moreover, existing SDP characterizations for frequency-selective SIP (FS-SIP) do not allow to formulate synthesis problems as SDPs. In this paper, we propose a completely new SDP characterization of general FS-SIP involving SDPs of moderate size and free from Lyapunov variables. Furthermore, a systematic IIR filter and filter bank design is developed in a similar vein, with several simulations provided to validate the effectiveness of our SDP formulation.
We consider a recent branch-and-bound algorithm of the authors for nonconvex quadratic programming. The algorithm is characterized by its use of semidefinite relaxations within a finite branching scheme. In this paper...
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We consider a recent branch-and-bound algorithm of the authors for nonconvex quadratic programming. The algorithm is characterized by its use of semidefinite relaxations within a finite branching scheme. In this paper, we specialize the algorithm to the box-constrained case and study its implementation, which is shown to be a state-of-the-art method for globally solving box-constrained nonconvex quadratic programs.
Hit-and-run algorithms are Monte Carlo methods for detecting necessary constraints in convex programming including semidefinite programming. The well known of these in semidefinite programming are semidefinite coordin...
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Hit-and-run algorithms are Monte Carlo methods for detecting necessary constraints in convex programming including semidefinite programming. The well known of these in semidefinite programming are semidefinite coordinate directions (SCD), semidefinite hypersphere directions (SHD) and semidefinite stand-and-hit (SSH) algorithms. SCD is considered to be the best on average and hence we use it for comparison. We develop two new hit-and-run algorithms in semidefinite programming that use diagonal directions. They are the uniform semidefinite diagonal directions (uniform SDD) and the original semidefinite diagonal directions (original SDD) algorithms. We analyze the costs and benefits of this change in comparison with SCD. We also show that both uniform SDD and original SDD generate points that are asymptotically uniform in the interior of the feasible region defined by the constraints. (C) 2008 Elsevier B.V. All rights reserved.
Given a basic compact semialgebraic set K subset of R(n), we introduce a methodology that generates a sequence converging to the volume of K. This sequence is obtained from optimal values of a hierarchy of either semi...
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Given a basic compact semialgebraic set K subset of R(n), we introduce a methodology that generates a sequence converging to the volume of K. This sequence is obtained from optimal values of a hierarchy of either semidefinite or linear programs. Not only the volume but also every finite vector of moments of the probability measure that is uniformly distributed on K can be approximated as closely as desired, which permits the approximation of the integral on K of any given polynomial;the extension to integration against some weight functions is also provided. Finally, some numerical issues associated with the algorithms involved are briefly discussed.
We present an analytic center cutting surface algorithm that uses mixed linear and multiple second-order cone cuts. Theoretical issues and applications of this technique are discussed. From the theoretical viewpoint, ...
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We present an analytic center cutting surface algorithm that uses mixed linear and multiple second-order cone cuts. Theoretical issues and applications of this technique are discussed. From the theoretical viewpoint, we derive two complexity results. We show that an approximate analytic center can be recovered after simultaneously adding p second-order cone cuts in O(plog (p+1)) Newton steps, and that the overall algorithm is polynomial. From the application viewpoint, we implement our algorithm on mixed linear-quadratic-semidefinite programming problems with bounded feasible region and report some computational results on randomly generated fully dense problems. We compare our CPU time with that of SDPLR, SDPT3, and SeDuMi and show that our algorithm outperforms these software packages on problems with fully dense coefficient matrices. We also show the performance of our algorithm on semidefinite relaxations of the maxcut and Lovasz theta problems.
Considering an infinite number of eigenvalues for time delay systems, it is difficult to determine their stability. We have developed a new approach for the stability test of time delay nonlinear hybrid systems. Const...
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Considering an infinite number of eigenvalues for time delay systems, it is difficult to determine their stability. We have developed a new approach for the stability test of time delay nonlinear hybrid systems. Construction of Lyapunov functions for hybrid systems is generally a difficult task, but once these functions are found, stability's analysis of the system is straight-Forward. In this paper both delay-independent and delay-dependent stability tests arc proposed, based oil the construction of appropriate Lyapunov-Krasovskii functionals. The methodology is based oil the sum of squares decomposition of multivariate polynomials and the algorithmic construction is achieved through the use of semidefinite programming. The reduction techniques provide numerical Solution Of large-scale instances;otherwise they will be computationally infeasible to solve. The introduced method can be used for hybrid systems with linear or nonlinear vector fields. Finally simulation results show the correctness and validity of the designed method.
We consider semidefinite programming (SDP) formulations of certain truss topology optimization problems, where a lower bound is imposed on the fundamental frequency of vibration of the truss structure. These SDP formu...
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We consider semidefinite programming (SDP) formulations of certain truss topology optimization problems, where a lower bound is imposed on the fundamental frequency of vibration of the truss structure. These SDP formulations were introduced in Ohsaki et al. (Comp. Meth. Appl. Mech. Eng. 180:203-217, 1999). We show how one may automatically obtain symmetric designs, by eliminating the 'redundant' symmetry in the SDP problem formulation. This has the advantage that the original SDP problem is substantially reduced in size for trusses with large symmetry groups.
We consider the nonconvex problem (RQ) of minimizing the ratio of two nonconvex quadratic functions over a possibly degenerate ellipsoid. This formulation is motivated by the so-called regularized total least squares ...
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We consider the nonconvex problem (RQ) of minimizing the ratio of two nonconvex quadratic functions over a possibly degenerate ellipsoid. This formulation is motivated by the so-called regularized total least squares problem (RTLS), which is a special case of the problem's class we study. We prove that under a certain mild assumption on the problem's data, problem (RQ) admits an exact semidefinite programming relaxation. We then study a simple iterative procedure which is proven to converge superlinearly to a global solution of (RQ) and show that the dependency of the number of iterations on the optimality tolerance epsilon grows as O(root ln epsilon(-1)).
For orthogonal frequency-division multiplexing (OFDM) systems equipped with multiple receive antennas, conventional pre-discrete Fourier transform (DFT) combining cannot balance the signal-to-noise ratio (SNR) perform...
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For orthogonal frequency-division multiplexing (OFDM) systems equipped with multiple receive antennas, conventional pre-discrete Fourier transform (DFT) combining cannot balance the signal-to-noise ratio (SNR) performance of all subcarriers. This degrades the whole system performance. In this paper, we propose the use of the max-min fair criterion for pre-DFT combining to solve the problem. semidefinite relaxation (SDR) is employed to approximate the solution. We also present a simple way to effectively reduce the amount of computation without sacrificing much of the performance. Simulation results show that, for both uncoded and coded OFDM systems, the proposed max-min fair pre-DFT combining solved via SDR can outperform conventional pre-DFT combining under various multipath channel environments with affordable computational complexity.
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