Although the lift-and-project operators of Lovasz and Schrijver have been the subject of intense study, their M(K, K) operator has received little attention. We consider an application of this operator to the stable s...
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Although the lift-and-project operators of Lovasz and Schrijver have been the subject of intense study, their M(K, K) operator has received little attention. We consider an application of this operator to the stable set problem. We begin with an initial linear programming (LP) relaxation consisting of clique and non-negativity inequalities, and then apply the operator to obtain a stronger extended LP relaxation. We discuss theoretical properties of the resulting relaxation, describe the issues that must be overcome to obtain an effective practical implementation, and give extensive computational results. Remarkably, the upper bounds obtained are sometimes stronger than those obtained with semidefinite programming techniques.
Given a compact basic semi-algebraic set K subset of R-n, a rational fraction f : R-n -> R, and a sequence of scalars y = (y(alpha)), we investigate when y(alpha) = integral K x(alpha) f d mu holds for all alpha is...
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Given a compact basic semi-algebraic set K subset of R-n, a rational fraction f : R-n -> R, and a sequence of scalars y = (y(alpha)), we investigate when y(alpha) = integral K x(alpha) f d mu holds for all alpha is an element of N-n, i.e., when y is the moment sequence of some measure f d mu, supported on K. This yields a set of (convex) linear matrix inequalities(LMI). We also use semidefinite programming to develop a converging approximation scheme to evaluate the integral integral f d mu when the moments of mu are known and f is a given rational fraction. Numerical expreriments are also provided. We finally provide (again LMI) conditions on the moments of two measures v, mu with support contained in K, to have dv = f d mu for some rational fraction f.
In this paper, we present a discrete-time optimization framework for target tracking with multi-agent systems. The "target tracking" problem is formulated as a generic semidefinite program (SDP) that when pa...
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In this paper, we present a discrete-time optimization framework for target tracking with multi-agent systems. The "target tracking" problem is formulated as a generic semidefinite program (SDP) that when paired with an appropriate objective yields an optimal robot configuration over a given time step. The framework affords impressive performance guarantees to include full target coverage (i.e. each target is tracked by at least a single team member) as well as maintenance of network connectivity across the formation. Key to this work is the result from spectral graph theory that states the second-smallest eigenvalue-lambda (2)-of a weighted graph's Laplacian (i.e. its inter-connectivity matrix) is a measure of connectivity for the associated graph. Our approach allows us to articulate agent-target coverage and inter-agent communication constraints as linear-matrix inequalities (LMIs). Additionally, we present two key extensions to the framework by considering alternate tracking problem formulations. The first allows us to guarantee k-coverage of targets, where each target is tracked by k or more agents. In the second, we consider a relaxed formulation for the case when network connectivity constraints are superfluous. The problem is modeled as a second-order cone program (SOCP) that can be solved significantly more efficiently than its SDP counterpart-making it suitable for large-scale teams (e.g. 100's of nodes in real-time). Methods for enforcing inter-agent proximity constraints for collision avoidance are also presented as well as simulation results for multi-agent systems tracking mobile targets in both a"e(2) and a"e(3).
The knowledge of the channel at the transmit side of a communication system can be exploited by using precoding techniques, from which the overall transmission quality might benefit significantly. However, in practica...
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The knowledge of the channel at the transmit side of a communication system can be exploited by using precoding techniques, from which the overall transmission quality might benefit significantly. However, in practical wireless systems, the channel state information is prone to errors, which sometimes deteriorates the performance drastically. In this paper, we address the problem of robust transceiver design in a downlink multiuser system, with respect to the erroneous channel knowledge at the transmitter. The base station is equipped with an antenna array, while users have single antennas. The transceiver optimization is performed under a set of predefined users' quality-of-service constraints, defined as maximum mean square errors, or minimum signal-to-interference-plus-noise ratios (SINRs), which must be satisfied for all disturbances that belong to given, bounded uncertainty sets. Efficient numerical solutions are obtained using semidefinite programming methods from convex optimization theory. The proposed algorithms are found to outperform related approaches in the literature in terms of the achieved performance, while maintaining low computational complexity. The studied uncertainty models are applicable in mitigating typical errors that emerge as a result of quantization or channel estimation.
We study robust transceiver optimization in a downlink, multiuser, wireless system, where the transmitter and the receivers are equipped with antenna arrays. The robustness is defined with respect to imperfect knowled...
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We study robust transceiver optimization in a downlink, multiuser, wireless system, where the transmitter and the receivers are equipped with antenna arrays. The robustness is defined with respect to imperfect knowledge of the channel at the transmitter. The errors in the channel state information are assumed to be bounded, and certain quality-of-service targets in terms of mean-square errors (MSEs) are guaranteed for all channels from the uncertainty regions. Iterative algorithms are proposed for the transceiver design. The iterations perform alternating optimization of the transmitter and the receivers and have equivalent semidefinite programming representations with efficient numerical solutions. The framework supports robust counterparts of several MSE-optimization problems, including transmit power minimization with per-user or per-stream MSE constraints, sum MSE minimization, min-max fairness, etc. Although the convergence to the global optimum cannot be claimed due to the intricacy of the problems, numerical examples show good practical performance of the presented methods. We also provide various possibilities for extensions in order to accommodate a broader set of scenarios regarding the precoder structure, the uncertainty modeling, and a multicellular setup.
The nuclear norm (sum of singular values) of a matrix is often used in convex heuristics for rank minimization problems in control, signal processing, and statistics. Such heuristics can be viewed as extensions of l(1...
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The nuclear norm (sum of singular values) of a matrix is often used in convex heuristics for rank minimization problems in control, signal processing, and statistics. Such heuristics can be viewed as extensions of l(1)-norm minimization techniques for cardinality minimization and sparse signal estimation. In this paper we consider the problem of minimizing the nuclear norm of an affine matrix-valued function. This problem can be formulated as a semidefinite program, but the reformulation requires large auxiliary matrix variables, and is expensive to solve by general-purpose interior-point solvers. We show that problem structure in the semidefinite programming formulation can be exploited to develop more efficient implementations of interior-point methods. In the fast implementation, the cost per iteration is reduced to a quartic function of the problem dimensions and is comparable to the cost of solving the approximation problem in the Frobenius norm. In the second part of the paper, the nuclear norm approximation algorithm is applied to system identification. A variant of a simple subspace algorithm is presented in which low-rank matrix approximations are computed via nuclear norm minimization instead of the singular value decomposition. This has the important advantage of preserving linear matrix structure in the low-rank approximation. The method is shown to perform well on publicly available benchmark data.
In this paper we consider the standard linear SDP problem, and its low rank nonlinear programming reformulation, based on a Gramian representation of a positive semidefinite matrix. For this nonconvex quadratic proble...
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In this paper we consider the standard linear SDP problem, and its low rank nonlinear programming reformulation, based on a Gramian representation of a positive semidefinite matrix. For this nonconvex quadratic problem with quadratic equality constraints, we give necessary and sufficient conditions of global optimality expressed in terms of the Lagrangian function.
We review several (and provide new) results on the theory of moments, sums of squares, and basic semialgebraic sets when convexity is present. In particular, we show that, under convexity, the hierarchy of semidefinit...
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We review several (and provide new) results on the theory of moments, sums of squares, and basic semialgebraic sets when convexity is present. In particular, we show that, under convexity, the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence, a highly desirable feature as convex problems are in principle easier to solve. In addition, if a basic semialgebraic set K is convex but its defining polynomials are not, we provide two algebraic certificates of convexity which can be checked numerically. The second is simpler and holds if a sufficient (and almost necessary) condition is satisfied;it also provides a new condition for K to have semidefinite representation. For this we use (and extend) some of the recent results from the author and Helton and Nie [Math. Program., to appear]. Finally, we show that, when restricting to a certain class of convex polynomials, the celebrated Jensen's inequality in convex analysis can be extended to linear functionals that are not necessarily probability measures.
We consider an ad hoc wireless network consisting of d source-destination pairs communicating, in a pairwise manner, via R relaying nodes. The relay nodes wish to cooperate, through a decentralized beamforming algorit...
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We consider an ad hoc wireless network consisting of d source-destination pairs communicating, in a pairwise manner, via R relaying nodes. The relay nodes wish to cooperate, through a decentralized beamforming algorithm, in order to establish all the communication links from each source to its respective destination. Our communication strategy consists of two steps. In the first step, all sources transmit their signals simultaneously. As a result, each relay receives a noisy faded mixture of all source signals. In the second step, each relay transmits an amplitude-and phase-adjusted version of its received signal. That is each relay multiply its received signal by a complex coefficient and retransmits the so-obtained signal. Our goal is to obtain these complex coefficients (beamforming weights) through minimization of the total relay transmit power while the signal-to-interference-plus-noise ratio (SINR) at the destinations are guaranteed to be above certain pre-defined thresholds. Although such a power minimization problem is not convex, we use semidefinite relaxation to turn this problem into a semidefinite programming (SDP) problem. Therefore, we can efficiently solve the SDP problem using interior point methods. Our numerical examples reveal that for high network data rates, our space division multiplexing scheme requires significantly less total relay transmit power compared to other orthogonal multiplexing schemes, such as time-division multiple access schemes.
The goal of this paper is to formulate and solve free material optimization problems with constraints on the smallest eigenfrequency of the optimal structure. A natural formulation of this problem as a linear semidefi...
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The goal of this paper is to formulate and solve free material optimization problems with constraints on the smallest eigenfrequency of the optimal structure. A natural formulation of this problem as a linear semidefinite program turns out to be numerically intractable. As an alternative, we propose a new approach, which is based on a nonlinear semidefinite low-rank approximation of the semidefinite dual. We introduce an algorithm based on this approach and analyze its convergence properties. The article is concluded by numerical experiments proving the effectiveness of the new approach.
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