We present a unifying framework to establish a lower bound on the number of semidefinite-programming-based lift-and-project iterations (rank) for computing the convex hull of the feasible solutions of various combinat...
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We present a unifying framework to establish a lower bound on the number of semidefinite-programming-based lift-and-project iterations (rank) for computing the convex hull of the feasible solutions of various combinatorial optimization problems. This framework is based on the maps which are commutative with the lift-and-project operators. Some special commutative maps were originally observed by Lovasz and Schrijver and have been used usually implicitly in the previous lower-bound analyses. In this paper, we formalize the lift-and-project commutative maps and propose a general framework for lower-bound analysis, in which we can recapture many of the previous lower-bound results on the lift-and-project ranks. (c) 2007 Elsevier B.V. All rights reserved.
Many combinatorial optimization problems can be modelled as polynomial-programming problems in binary variables that are all 0-1 or 1. A sufficient condition under which a common method for obtaining semidefinite-prog...
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Many combinatorial optimization problems can be modelled as polynomial-programming problems in binary variables that are all 0-1 or 1. A sufficient condition under which a common method for obtaining semidefinite-programming relaxations of the two models of the same problem gives equivalent relaxations is established. (C) 2007 Elsevier B.V. All rights reserved.
Given a sample covariance matrix, we solve a maximum likelihood problem penalized by the number of nonzero coefficients in the inverse covariance matrix. Our objective is to find a sparse representation of the sample ...
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Given a sample covariance matrix, we solve a maximum likelihood problem penalized by the number of nonzero coefficients in the inverse covariance matrix. Our objective is to find a sparse representation of the sample data and to highlight conditional independence relationships between the sample variables. We first formulate a convex relaxation of this combinatorial problem, we then detail two efficient first-order algorithms with low memory requirements to solve large-scale, dense problem instances.
We consider the class of nonlinear optimal control problems (OCPs) with polynomial data, i.e., the differential equation, state and control constraints, and cost are all described by polynomials, and more generally fo...
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We consider the class of nonlinear optimal control problems (OCPs) with polynomial data, i.e., the differential equation, state and control constraints, and cost are all described by polynomials, and more generally for OCPs with smooth data. In addition, state constraints as well as state and/or action constraints are allowed. We provide a simple hierarchy of LMI- (linear matrix inequality)-relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value. Under some convexity assumptions, the sequence converges to the optimal value of the OCP. Preliminary results show that good approximations are obtained with few moments.
In this paper, the problem of distributed beam-forming is considered for a wireless network which consists of a transmitter, a receiver, and r relay nodes. For such a network, assuming that the second-order statistics...
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In this paper, the problem of distributed beam-forming is considered for a wireless network which consists of a transmitter, a receiver, and r relay nodes. For such a network, assuming that the second-order statistics of the channel coefficients are available, we study two different beamforming design approaches. As the first approach, we design the beamformer through minimization of the total transmit power subject to the receiver quality of service constraint. We show that this approach yields a closed-form solution. In the second approach, the beamforming weights are obtained through maximizing the receiver signal-to-noise ratio (SNR) subject to two different types of power constraints, namely the total transmit power constraint and individual relay power constraints. We show that the total power constraint leads to a closed-form solution while the individual relay power constraints result in a quadratic programming optimization problem. The later optimization problem does not have a closed-form solution. However, it is shown that using semidefinite relaxation, this problem can be turned into a convex feasibility semidefinite programming (SDP), and therefore, can be efficiently solved using interior point methods. Furthermore, we develop a simplified, thus suboptimal, technique which is computationally more efficient than the SDP approach. In fact, the simplified algorithm provides the beamforming weight vector in a closed form. Our numerical examples show that as the uncertainty in the channel state information is increased, satisfying the quality of service constraint becomes harder, i.e., it takes more power to satisfy these constraints. Also our simulation results show that when compared to the SDP-based method, our simplified technique suffers a 2-dB loss in SNR for low to moderate values of transmit power.
We develop a nonlinear minimax estimator for the classical linear regression model assuming that the true parameter vector lies in an intersection of ellipsoids. We seek an estimate that minimizes the worst-case estim...
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We develop a nonlinear minimax estimator for the classical linear regression model assuming that the true parameter vector lies in an intersection of ellipsoids. We seek an estimate that minimizes the worst-case estimation error over the given parameter set. Since this problem is intractable, we approximate it using semidefinite relaxation, and refer to the resulting estimate as the relaxed Chebyshev center (RCC). We show that the RCC is unique and feasible, meaning it is consistent with the prior information. We then prove that the constrained least-squares (CLS) estimate for this problem can also be obtained as a relaxation of the Chebyshev center, that is looser than the RCC. Finally, we demonstrate through simulations that the RCC can significantly improve the estimation error over the CLS method.
Modeling experiments with qualitative and quantitative factors is an important issue in computer modeling. We propose a framework for building Gaussian process model,, that incorporate both types of factors. The key t...
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Modeling experiments with qualitative and quantitative factors is an important issue in computer modeling. We propose a framework for building Gaussian process model,, that incorporate both types of factors. The key to the development of these new models is an approach for constructing correlation functions with qualitative and quantitative factors. An iterative estimation procedure is developed for the proposed models. Modern optimization techniques are used in the estimation to ensure the validity of the constructed correlation functions. The proposed method is illustrated with in example involving a known function and a real example for modeling the thermal distribution of a data center.
We investigate hierarchies of semidefinite approximations for the chromatic number chi(G) of a graph G. We introduce an operator Psi mapping any graph parameter beta(G), nested between the stability number alpha(G) an...
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We investigate hierarchies of semidefinite approximations for the chromatic number chi(G) of a graph G. We introduce an operator Psi mapping any graph parameter beta(G), nested between the stability number alpha(G) and chi((G) over bar), to a new graph parameter Psi(beta)(G), nested between alpha((G) over bar) and chi(G);Psi(beta)(G) is polynomial time computable if beta(G) is. As an application, there is no polynomial time computable graph parameter nested between the fractional chromatic number chi*(.) and chi(.) unless P = NP. Moreover, based on the Motzkin-Straus formulation for alpha(G), we give (quadratically constrained) quadratic and copositive programming formulations for chi(G). Under some mild assumptions, n/beta(G) <= Psi(beta)(G), but, while n/beta(G) remains below chi*(G), Psi(beta)(G) can reach chi(G) (e. g., for beta(.) = alpha(.)). We also define new polynomial time computable lower bounds for chi(G), improving the classic Lovasz theta number (and its strengthenings obtained by adding nonnegativity and triangle inequalities);experimental results on Hamming graphs, Kneser graphs, and DIMACS benchmark graphs will be given in the follow-up paper [N. Gvozdenovic and M. Laurent, SIAM J. Optim., 19 (2008), pp. 592-615].
The problem of computing approximate GCDs of several polynomials with real or complex coefficients can be formulated as computing the minimal perturbation such that the perturbed polynomials have an exact GCD of given...
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The problem of computing approximate GCDs of several polynomials with real or complex coefficients can be formulated as computing the minimal perturbation such that the perturbed polynomials have an exact GCD of given degree. We present algorithms based on SOS (Sums Of Squares) relaxations for solving the involved polynomial or rational function optimization problems with or without constraints. (c) 2008 Elsevier B.V. All rights reserved.
Many object recognition and localization techniques utilize multiple levels of local representations. These local feature representations are common, and one way to improve the efficiency of algorithms that use them i...
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Many object recognition and localization techniques utilize multiple levels of local representations. These local feature representations are common, and one way to improve the efficiency of algorithms that use them is to reduce the size of the local representations. There has been previous work on selecting Subsets of image features, but the focus here is on a systematic study of the feature selection problem. We have developed a combinatorial characterization of the feature subset selection problem that leads to a general optimization framework. This framework optimizes multiple objectives and allows the encoding of global constraints. The features selected by this algorithm are able to achieve improved performance on the problem of object localization. We present a dataset of synthetic images, along with ground-truth information, which allows us to precisely measure and compare the performance of feature subset algorithms. Our experiments show that Subsets of image features produced by our method, stable bounded canonical sets (SBCS), outperform subsets produced by K-Means clustering and threshold-based methods for the task of object localization under occlusion. (C) 2008 Elsevier Inc. All rights reserved.
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