This paper proposes a new method for designing a class of two-channel perfect reconstruction (PR) linear-phase FIR filterbanks (FBs) and wavelets previously proposed by Phoong et al. By expressing the given K-regulari...
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This paper proposes a new method for designing a class of two-channel perfect reconstruction (PR) linear-phase FIR filterbanks (FBs) and wavelets previously proposed by Phoong et al. By expressing the given K-regularity constraints as a set of linear equality constraints in the design variables, the design problem using the minimax error criterion can be solved using semidefinite programming (SDP). Design examples show that the proposed method is very effective and it yields equiripple stopband response while satisfying the given K-regularity condition.
We consider 3-partitioning the vertices of a graph into sets S-1, S-2, and S-3 of specified cardinalities, such that the total weight of all edges joining S1 and S2 is minimized. This problem is closely related to sev...
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We consider 3-partitioning the vertices of a graph into sets S-1, S-2, and S-3 of specified cardinalities, such that the total weight of all edges joining S1 and S2 is minimized. This problem is closely related to several NP-hard problems like determining the bandwidth or finding a vertex separator in a graph. We show that this problem can be formulated as a linear program over the cone of completely positive matrices, leading in a natural way to semidefinite relaxations of the problem. We show in particular that the spectral relaxation introduced by Helmberg et al. (1995) can equivalently be formulated as a semidefinite program. Finally we propose a tightened version of this semidefinite program and show on some small instances that this new bound is a significant improvement over the spectral bound.
Given any open convex cone K, a logarithmically homogeneous, self-concordant barrier for K, and any positive real number r < 1, we associate, with each direction x is an element of K, a second-order cone (K) over c...
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Given any open convex cone K, a logarithmically homogeneous, self-concordant barrier for K, and any positive real number r < 1, we associate, with each direction x is an element of K, a second-order cone (K) over cap (r)(x) containing K. We show that K is the interior of the intersection of the second-order cones (K) over cap (r)(x), as x ranges over all directions in K. Using these second-order cones as approximations to cones of symmetric, positive definite matrices, we develop a new polynomial-time primal-dual interior-point algorithm for semidefinite programming. The algorithm is extended to symmetric cone programming via the relation between symmetric cones and Euclidean Jordan algebras.
The falling price and reduced size of sensors for monitoring spatially-sensitive environmental properties such as temperature, light, sound, and vibration have motivated research in location algorithms in recent years...
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The falling price and reduced size of sensors for monitoring spatially-sensitive environmental properties such as temperature, light, sound, and vibration have motivated research in location algorithms in recent years. To our knowledge, the algorithm which achieves the best performance refines erroneous measurements through an optimization program whose quadratic constraints force the sensors to be consistent with the geometry of the physical world. Since the program is non-convex, the authors relax the constraints to render it convex for which efficient solution methods exist. We propose solving a similar optimization program however by applying convex geometrical constraints directly, necessitating no relaxation of the constraints and in turn ensuring a solution still compliant with the physical world. We show through extensive experimentation that ours outperforms the competing algorithm across all network parameters. In addition, this paper formulates a distributed version of our algorithm which achieves the same globally optimal objective function as the centralized version, and reports the messaging overhead for its convergence.
We give a complete characterization of constant quadratic functions over an affine variety. This result is used to convexify the objective function of a general quadratic programming problem (Pb) which contains linear...
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We give a complete characterization of constant quadratic functions over an affine variety. This result is used to convexify the objective function of a general quadratic programming problem (Pb) which contains linear equality constraints. Thanks to this convexification, we show that one can express as a semidefinite program the dual of the partial Lagrangian relaxation of (Pb) where the linear constraints are not relaxed. We apply these results by comparing two semidefinite relaxations made from two sets of null quadratic functions over an affine variety.
A number of recent papers on approximation algorithms have used the square roots of unity, -1 and 1, to represent binary decision variables for problems in combinatorial optimization, and have relaxed these to unit ve...
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ISBN:
(纸本)9781581133493
A number of recent papers on approximation algorithms have used the square roots of unity, -1 and 1, to represent binary decision variables for problems in combinatorial optimization, and have relaxed these to unit vectors in real space using semidefinite programming in order to obtain near optimum solutions to these problems. In this paper, we consider using the cube roots of unity, 1, e(i2pi/3), and e(i4pi/3), to represent ternary decision variables for problems in combinatorial optimization. Here the natural relaxation is that of unit vectors in complex space. We use an extension of semidefinite programming to complex space to solve the natural relaxation, and use a natural extension of the random hyperplane technique introduced by the authors in Goemans and Williamson (J. ACM 42 (1995) 1115-1145) to obtain near-optimum solutions to the problems. In particular, we consider the problem of maximizing the total weight of satisfied equations x(u) - x(v) drop c (mod 3) and inequations x - x(v) not equivalent to c (mod 3), where x(u) epsilon {0, 1, 2} for all u. This problem can be used to model the MAx-3-CUT problem and a directed variant we call MAX-3-DICUT. For the general problem, we obtain a 0.793733-approximation algorithm. If the instance contains only inequations (as it does for MAX-3-CUT), we obtain a performance guarantee of (7)/(12) + (3)/(4pi2) arccos(2)(- 1/4) - epsilon>0.836008. This compares with proven performance guarantees of 0.800217 for MAX-3-CUT (by Frieze and Jerrum (Algorithmica 18 (1997) 67-81) and 1 + 10(-8) for the general problem (by Andersson et al. (J. Algorithms 3 39 (2001) 162-204)). It matches the guarantee of 0.836008 for MAX-3-CUT found independently by de Klerk et al. (On approximate graph colouring and Max-k-Cut algorithms based on the 9-function, Manuscript, October 2000). We show that all these algorithms are in fact equivalent in the case of MAX-3CUT, and that our algorithm is the same as that of Andersson et al. in the more gener
A successive quadratic programming algorithm for solving SDP relaxation of Max- Bisection is provided and its convergence result is given. The step-size in the algorithm is obtained by solving n easy quadratic equatio...
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A successive quadratic programming algorithm for solving SDP relaxation of Max- Bisection is provided and its convergence result is given. The step-size in the algorithm is obtained by solving n easy quadratic equations without using the linear search technique. The numerical experiments show that this algorithm is rather faster than the interior-point method.
We discuss a method for multidimensional FIR filter design via sum-of-squares formulations of spectral mask constraints. The sum-of-squares optimization problem is expressed as a semidefinite program with low-rank str...
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We discuss a method for multidimensional FIR filter design via sum-of-squares formulations of spectral mask constraints. The sum-of-squares optimization problem is expressed as a semidefinite program with low-rank structure, by sampling the constraints using discrete cosine and sine transforms. The resulting semidefinite program is then solved by a customized primal-dual interior-point method that exploits low-rank structure. This leads to a substantial reduction in the computational complexity, compared to general-purpose semidefinite programming methods that exploit sparsity.
We present an algorithm that determines whether an n-port with given impedance matrix at a set of frequencies can be realised from ideal transformers and multiple instances of a set of linear, time-invariant component...
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We present an algorithm that determines whether an n-port with given impedance matrix at a set of frequencies can be realised from ideal transformers and multiple instances of a set of linear, time-invariant components (specified by their admittance parameters). The algorithm is based on solving a linear matrix inequality derived using Tellegen's theorem. If the n-port can be realised, the noise figure can be minimized by a choice of cost function that is described. We use as an example an amplifier using realistic RF CMOS components.
Supervised subspace learning techniques have been extensively studied in biometrics literature;however, there is little work dedicated to: 1) how to automatically determine the subspace dimension in the context of sup...
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Supervised subspace learning techniques have been extensively studied in biometrics literature;however, there is little work dedicated to: 1) how to automatically determine the subspace dimension in the context of supervised learning, and 2) how to explicitly guarantee the classification performance on a training set. In this paper, by following our previous work on unified subspace learning framework in our earlier work, we present a general framework, called parameter-free graph embedding (PFGE) to solve the above two problems by posing a general supervised subspace learning task as a semidefinite programming problem. The semipositive feature Gram matrix, namely the product of the transformation matrix and its transpose, is derived by optimizing a trace difference form of an objective function extended from that in our earlier work with the constraints that guarantee the class homogeneity within the neighborhood of each datum. Then, the subspace dimension and the feature weights are simultaneously obtained via the singular value decomposition of the feature Gram matrix. In addition, to alleviate the computational complexity, the Kronecker product approximation of the feature Gram matrix is proposed by taking advantage of the essential matrix form of image pixels. The experiments on simulated data and real-world data demonstrate the capability of the new PFGE framework in estimating the subspace dimension for supervised learning as well as the superiority in classification performance over traditional algorithms for subspace learning.
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