The present paper points out that a class of positive polynomials deserves special attention due to several interesting applications in signal processing, system analysis and control. We consider positive hybrid polyn...
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The present paper points out that a class of positive polynomials deserves special attention due to several interesting applications in signal processing, system analysis and control. We consider positive hybrid polynomials with two variables, one real, the other complex, belonging to the unit circle. We present several theoretical results regarding the sum-of-squares representations of such polynomials, treating the cases where positivity occurs globally or on domains. We also give a specific Bounded Real Lemma. All the characterizations of positive hybrid polynomials are expressed in terms of positive semidefinite matrices and can be extended to polynomials with more than two variables. On the applicative side, we show how several problems are numerically tractable via semidefinite programming (SDP) algorithms. The first problem is the minimax design of adjustable FIR filters, using a modified Farrow structure. We discuss linear-phase and approximately linear-phase designs. The second is the absolute stability of time-delay feedback systems with unknown delay, for which we treat the cases of bounded and unbounded delay. Finally, we discuss the application of our methods to checking the stability of parameter-dependent systems. The design procedures are illustrated with numerical examples.
Distance metric learning and classifier design are two highly challenging tasks in the machine learning community. In this paper we propose a new large margin nearest local mean (LMNLM) scheme to consider them jointly...
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Distance metric learning and classifier design are two highly challenging tasks in the machine learning community. In this paper we propose a new large margin nearest local mean (LMNLM) scheme to consider them jointly, which aims at improving the separability between local parts of different classes. We adopt 'local mean vector' as the basic classification model, and then through linear transformation, large margins between heterogeneous local parts are introduced. Moreover, by eigenvalue decomposition, we may also reduce data's dimensions. LMNLM can be formulated as a semidefinite programming (SDP) problem, so it is assured to converge globally. Experimental results show that LMNLM is a promising algorithm due to its leading to high classification accuracies and low dimensions. (C) 2009 Elsevier B.V. All rights reserved.
The second smallest eigenvalue of the Laplace matrix of a graph and its eigenvectors, also known as Fiedler vectors in spectral graph partitioning, carry significant structural information regarding the connectivity o...
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The second smallest eigenvalue of the Laplace matrix of a graph and its eigenvectors, also known as Fiedler vectors in spectral graph partitioning, carry significant structural information regarding the connectivity of the graph. Using semidefinite programming duality, we offer a geometric interpretation of this eigenspace as optimal solution to a graph realization problem. A corresponding interpretation is also given for the eigenspace of the maximum eigenvalue of the Laplacian. (C) 2010 Elsevier B.V. All rights reserved.
The main goal of this paper is to develop a numerical procedure for construction of covariance matrices such that for a given covariance structural model and a discrepancy function the corresponding minimizer of the d...
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The main goal of this paper is to develop a numerical procedure for construction of covariance matrices such that for a given covariance structural model and a discrepancy function the corresponding minimizer of the discrepancy function has a specified value. Often construction of such matrices is a first step in Monte Carlo studies of statistical inferences of misspecified models. We analyze theoretical aspects of the problem and suggest a numerical procedure based on semidefinite programming techniques. As an example, we discuss in detail the factor analysis model.
This correspondence revisits the joint transceiver optimization problem for multiple-input multiple-output (MIMO) channels. The linear transceiver as well as the transceiver with linear precoding and decision feedback...
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This correspondence revisits the joint transceiver optimization problem for multiple-input multiple-output (MIMO) channels. The linear transceiver as well as the transceiver with linear precoding and decision feedback equalization are considered. For both types of transceivers, in addition to the usual total power constraint, an individual power constraint on each antenna element is also imposed. A number of objective functions including the average bit error rate, are considered for both of the above systems under the generalized power constraint. It is shown that for both types of systems the optimization problem can be solved by first solving a class of MMSE problems (AM-MMSE or GM-MMSE depending on the type of transceiver), and then using majorization theory. The first step, under the generalized power constraint, can be formulated as a semidefinite program (SDP) for both types of transceivers, and can be solved efficiently by convex optimization tools. The second step is addressed by using results from majorization theory. The framework developed here is general enough to add any finite number of linear constraints to the covariance matrix of the input.
We provide two certificates of convexity for arbitrary basic closed semi-algebraic sets of R(n). The first one is based on a necessary and sufficient condition whereas the second one is based on a sufficient (but simp...
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We provide two certificates of convexity for arbitrary basic closed semi-algebraic sets of R(n). The first one is based on a necessary and sufficient condition whereas the second one is based on a sufficient (but simpler) condition only. Both certificates are obtained from any feasible solution of a related semidefinite program and so, in principle, can be obtained numerically (however, up to machine precision). (c) 2010 Elsevier Ltd. All rights reserved.
It has recently been shown (Burer, Math Program 120: 479-495, 2009) that a large class of NP-hard nonconvex quadratic programs (NQPs) can be modeled as so-called completely positive programs, i.e., the minimization of...
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It has recently been shown (Burer, Math Program 120: 479-495, 2009) that a large class of NP-hard nonconvex quadratic programs (NQPs) can be modeled as so-called completely positive programs, i.e., the minimization of a linear function over the convex cone of completely positive matrices subject to linear constraints. Such convex programs are NP-hard in general. A basic tractable relaxation is gotten by approximating the completely positive matrices with doubly nonnegative matrices, i.e., matrices which are both nonnegative and positive semidefinite, resulting in a doubly nonnegative program (DNP). Optimizing a DNP, while polynomial, is expensive in practice for interior-point methods. In this paper, we propose a practically efficient decomposition technique, which approximately solves the DNPs while simultaneously producing lower bounds on the original NQP. We illustrate the effectiveness of our approach for solving the basic relaxation of box-constrained NQPs (BoxQPs) and the quadratic assignment problem. For one quadratic assignment instance, a best-known lower bound is obtained. We also incorporate the lower bounds within a branch-and-bound scheme for solving BoxQPs and the quadratic multiple knapsack problem. In particular, to the best of our knowledge, the resulting algorithm for globally solving BoxQPs is the most efficient to date.
We consider polynomial optimization problems pervaded by a sparsity pattern. It has been shown in Lasserre (SIAM J. Optim. 17(3):822-843, 2006) and Waki et al. (SIAM J. Optim. 17(1):218-248, 2006) that the optimal sol...
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We consider polynomial optimization problems pervaded by a sparsity pattern. It has been shown in Lasserre (SIAM J. Optim. 17(3):822-843, 2006) and Waki et al. (SIAM J. Optim. 17(1):218-248, 2006) that the optimal solution of a polynomial programming problem with structured sparsity can be computed by solving a series of semidefinite relaxations that possess the same kind of sparsity. We aim at solving the former relaxations with a decomposition-based method, which partitions the relaxations according to their sparsity pattern. The decomposition-based method that we propose is an extension to semidefinite programming of the Benders decomposition for linear programs (Benders, Comput. Manag. Sci. 2(1):3-19, 2005).
In this paper, vector variational inequalities (VVI) with matrix inequality constraints are investigated by using the image space analysis. Linear separation for VVI with matrix inequality constraints is characterized...
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In this paper, vector variational inequalities (VVI) with matrix inequality constraints are investigated by using the image space analysis. Linear separation for VVI with matrix inequality constraints is characterized by using the saddle-point conditions of the Lagrangian function. Lagrangian-type necessary and sufficient optimality conditions for VVI with matrix inequality constraints are derived by utilizing the separation theorem. Gap functions for VVI with matrix inequality constraints and weak sharp minimum property for the solutions set of VVI with matrix inequality constraints are also considered. The results obtained above are applied to investigate the Lagrangian-type necessary and sufficient optimality conditions for vector linear semidefinite programming problems as well as VVI with convex quadratic inequality constraints.
The facial reduction algorithm reduces the size of the positive semidefinite cone in SDP. The elimination method for a sparse SOS polynomial [M. Kojima, S. Kim, H. Waki, Sparsity in sums of squares of polynomials, Mat...
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The facial reduction algorithm reduces the size of the positive semidefinite cone in SDP. The elimination method for a sparse SOS polynomial [M. Kojima, S. Kim, H. Waki, Sparsity in sums of squares of polynomials, Math. Program. 103 (2005) 45-62] removes monomials which do not appear in any SOS representations. In this paper, we establish a relationship between a facial reduction algorithm and the elimination method for a sparse SOS polynomial. (C) 2010 Elsevier B.V. All rights reserved.
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