We propose a parameterization of the finite-impulse-response (FIR) filters that generate the higher density discrete wavelet transform (HD-DWT). The parameterization has the form of a linear system whose variables are...
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We propose a parameterization of the finite-impulse-response (FIR) filters that generate the higher density discrete wavelet transform (HD-DWT). The parameterization has the form of a linear system whose variables are the coefficients of two positive trigonometric polynomials. The parameterization allows the easy separation of independent variables and can be used to transform the optimization of HD-DWT into semidefinite programming (SDP) form. Using the parameterization, we are able to optimize filters whose length is greater than the minimal one imposed by regularity constraints. We also propose two optimization methods for the dual-tree HD-DWT. The first generalizes the design based on allpass approximation of the half sample delay. The second is a brute force optimization based on generating values for the independent variables by using successive sections through the admissible set. As optimization criteria, we use an analyticity measure in the latter method and stopband energy in the former ones. For each type of design, we present examples and compare them with previous work.
On the heels of compressed sensing, a new field has very recently emerged. This field addresses a broad range of problems of significant practical interest, namely, the recovery of a data matrix from what appears to b...
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On the heels of compressed sensing, a new field has very recently emerged. This field addresses a broad range of problems of significant practical interest, namely, the recovery of a data matrix from what appears to be incomplete, and perhaps even corrupted, information. In its simplest form, the problem is to recover a matrix from a small sample of its entries. It comes up in many areas of science and engineering, including collaborative filtering, machine learning, control, remote sensing, and computer vision, to name a few. This paper surveys the novel literature on matrix completion, which shows that under some suitable conditions, one can recover an unknown low-rank matrix from a nearly minimal set of entries by solving a simple convex optimization problem, namely, nuclear-norm minimization subject to data constraints. Further, this paper introduces novel results showing that matrix completion is provably accurate even when the few observed entries are corrupted with a small amount of noise. A typical result is that one can recover an unknown n x n matrix of low rank r from just about nr log(2)n noisy samples with an error that is proportional to the noise level. We present numerical results that complement our quantitative analysis and show that, in practice, nuclear-norm minimization accurately fills in the many missing entries of large low-rank matrices from just a few noisy samples. Some analogies between matrix completion and compressed sensing are discussed throughout.
Given an unconstrained quadratic optimization problem in the following form: (QP) min{x(t)Qx vertical bar x is an element of {-1,1}(n)}. with Q is an element of R-nxn, we present different methods for computing bounds...
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Given an unconstrained quadratic optimization problem in the following form: (QP) min{x(t)Qx vertical bar x is an element of {-1,1}(n)}. with Q is an element of R-nxn, we present different methods for computing bounds on its optimal objective value. Some of the lower bounds introduced are shown to generally improve over the one given by a classical semidefinite relaxation. We report on theoretical results on these new bounds and provide preliminary computational experiments on small instances of the maximum cut problem illustrating their performance. (C) 2010 Elsevier B.V. All rights reserved.
A numerical method for a (possibly nonconvex) scalar variational problem for the functional [image omitted] to be minimized where uW1, p() and u|=uD is proposed;n is a bounded Lipschitz domain, n=1 or 2. This method a...
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A numerical method for a (possibly nonconvex) scalar variational problem for the functional [image omitted] to be minimized where uW1, p() and u|=uD is proposed;< subset of>n is a bounded Lipschitz domain, n=1 or 2. This method allows the computation of the Young-measure solution of the generalized relaxed version of the original problem and applies to those cases in which phi 1(x, center dot) is polynomial. The Young measures involved in the relaxed problem can be represented by their algebraic moments, and a finite-element mesh is used to discretize and thus to approximate both u and the Young measure (in the momentum representation). Eventually, this obtained convex semidefinite program is solved by efficient specialized mathematical-programming solvers. This method is justified by convergence analysis and eventually tested on a 2-dimensional benchmark numerical example. It serves as an example of how convex compactification can efficiently be used numerically if osmallo enough, that is, coarse enough.
We consider one-round games between a classical verifier and two provers who share entanglement. We show that when the constraints enforced by the verifier are "unique" constraints (i.e., permutations), the ...
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We consider one-round games between a classical verifier and two provers who share entanglement. We show that when the constraints enforced by the verifier are "unique" constraints (i.e., permutations), the value of the game can be well approximated by a semidefinite program (SDP). Essentially the only algorithm known previously was for the special case of binary answers, as follows from the work of Tsirelson in 1980. Among other things, our result implies that the variant of the unique games conjecture where we allow the provers to share entanglement is false. Our proof is based on a novel "quantum rounding technique," showing how to take a solution to an SDP and transform it into a strategy for entangled provers. Using our approximation by an SDP, we also show a parallel repetition theorem for unique entangled games.
The classical narrow main beam and low sidelobe synthesis problem is addressed and extended to arbitrary sidelobe envelopes in both one and two dimensional scenarios. The proposed approach allows also to handle arbitr...
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The classical narrow main beam and low sidelobe synthesis problem is addressed and extended to arbitrary sidelobe envelopes in both one and two dimensional scenarios. The proposed approach allows also to handle arbitrary arrays. This extended basic synthesis problem is first rewritten as a convex optimization problem. This problem is then transformed into either a linear program or a second order cone program that can be solved efficiently by readily available software. The convex formulation is not strictly equivalent to the initial synthesis problem but arguments are given that establish that the obtained optimum is either identical or can be made as close as wanted to the desired one. Finally, numerical applications and a comparison with a classical solution are presented to both confirm the optimality of the solution and illustrate the potentialities of the proposed approach.
We present a method for finding exact solutions of Max-Cut, the problem of finding a cut of maximum weight in a weighted graph. We use a Branch-and-Bound setting that applies a dynamic version of the bundle method as ...
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We present a method for finding exact solutions of Max-Cut, the problem of finding a cut of maximum weight in a weighted graph. We use a Branch-and-Bound setting that applies a dynamic version of the bundle method as bounding procedure. This approach uses Lagrangian duality to obtain a "nearly optimal" solution of the basic semidefinite Max-Cut relaxation, strengthened by triangle inequalities. The expensive part of our bounding procedure is solving the basic semidefinite relaxation of the Max-Cut problem, which has to be done several times during the bounding process. We review other solution approaches and compare the numerical results with our method. We also extend our experiments to instances of unconstrained quadratic 0-1 optimization and to instances of the graph equipartition problem. The experiments show that our method nearly always outperforms all other approaches. In particular, for dense graphs, where linear programming-based methods fail, our method performs very well. Exact solutions are obtained in a reasonable time for any instance of size up to n = 100, independent of the density. For some problems of special structure we can solve even larger problem classes. We could prove optimality for several problems of the literature where, to the best of our knowledge, no other method is able to do so.
Finding global optimum of a non-convex quadratic function is in general a very difficult task even when the feasible set is a polyhedron. We show that when the feasible set of a quadratic problem consists of orthogona...
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Finding global optimum of a non-convex quadratic function is in general a very difficult task even when the feasible set is a polyhedron. We show that when the feasible set of a quadratic problem consists of orthogonal matrices from Rnxk, then we can transform it into a semidefinite program in matrices of order kn which has the same optimal value. This opens new possibilities to get good lower bounds for several problems from combinatorial optimization, like the Graph partitioning problem (GPP), the Quadratic assignment problem (QAP) etc. In particular we show how to improve significantly the well-known Donath-Hoffman eigenvalue lower bound for GPP by semidefinite programming. In the last part of the paper we show that the copositive strengthening of the semidefinite lower bounds for GPP and QAP yields the exact values.
The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields ...
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The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard because it contains vector cardinality minimization as a special case. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum-rank solution can be recovered by solving a convex optimization problem, namely, the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability, provided the codimension of the subspace is sufficiently large. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this preexisting concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. We also discuss several algorithmic approaches to minimizing the nuclear norm and illustrate our results with numerical examples.
The main objective in this work is to compare different convex relaxations for Model Predictive Control (MPC) problems with mixed real valued and binary valued control signals. In the problem description considered, t...
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The main objective in this work is to compare different convex relaxations for Model Predictive Control (MPC) problems with mixed real valued and binary valued control signals. In the problem description considered, the objective function is quadratic, the dynamics are linear, and the inequality constraints on states and control signals are all linear. The relaxations are related theoretically and the quality of the bounds and the computational complexities are compared in numerical experiments. The investigated relaxations include the Quadratic programming (QP) relaxation, the standard semidefinite programming (SDP) relaxation, and an equality constrained SDP relaxation. The equality constrained SDP relaxation appears to be new in the context of hybrid MPC and the result presented in this work indicates that it can be useful as an alternative relaxation, which is less computationally demanding than the ordinary SDP relaxation and which often gives a better bound than the bound from the QP relaxation. Furthermore, it is discussed how the result from the SDP relaxations can be used to generate suboptimal solutions to the control problem. Moreover, it is also shown that the equality constrained SDP relaxation is equivalent to a QP in an important special case. (c) 2010 Elsevier Ltd. All rights reserved.
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