Owing to their high accuracy and ease of formulation, there has been great interest in applying convex optimization techniques, particularly that of semidefinite programming (SDP) relaxation, to tackle the sensor netw...
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Owing to their high accuracy and ease of formulation, there has been great interest in applying convex optimization techniques, particularly that of semidefinite programming (SDP) relaxation, to tackle the sensor network localization problem in recent years. However, a drawback of such techniques is that the resulting convex program is often expensive to solve. In order to speed up computation, various edge sparsification heuristics have been proposed, whose aim is to reduce the number of edges in the input graph. Although these heuristics do reduce the size of the convex program and hence make it faster to solve, they are often ad hoc in nature and do not preserve the localization properties of the input. As such, one often has to face a tradeoff between solution accuracy and computational effort. In this paper, we propose a novel edge sparsification heuristic that can provably preserve the localization properties of the original input. At the heart of our heuristic is a graph decomposition procedure, which allows us to identify certain sparse generically universally rigid subgraphs of the input graph. Our computational results show that the proposed approach can significantly reduce the computational and memory complexities of SDP-based algorithms for solving the sensor network localization problem. Moreover, it compares favorably with existing speedup approaches, both in terms of accuracy and solution time.
This paper considers the break minimization problem in sports timetabling. The problem is to find, under a given timetable of a round-robin tournament, a home-away assignment that minimizes the number of breaks, i.e.,...
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This paper considers the break minimization problem in sports timetabling. The problem is to find, under a given timetable of a round-robin tournament, a home-away assignment that minimizes the number of breaks, i.e., the number of occurrences of consecutive matches held either both at away or both at home for a team. We formulate the break minimization problem as MAX RES CUT and MAX 2SAT, and apply Goemans and Williamson's approximation algorithm using semidefinite programming. Computational experiments show that our approach quickly generates solutions of good approximation ratios. (c) 2004 Elsevier Ltd. All rights reserved.
A sensor network localization problem is to determine the positions of the sensor nodes in a network given incomplete and inaccurate pairwise distance measurements. Such distance data may be acquired by a sensor node ...
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A sensor network localization problem is to determine the positions of the sensor nodes in a network given incomplete and inaccurate pairwise distance measurements. Such distance data may be acquired by a sensor node by communicating with its neighbors. We describe a general semidefinite programming (SDP)-based approach for solving the graph realization problem, of which the sensor network localization problems is a special case. We investigate the performance of this method on problems with noisy distance data. Error bounds are derived from the SDP formulation. The sources of estimation error in the SDP formulation are identified. The SDP solution usually has a rank higher than the underlying physical space which, when projected onto the lower dimensional space, generally results in high estimation error. We describe two improvements to ameliorate such a difficulty. First, we propose a regularization term in the objective function that can help to reduce the rank of the SDP solution. Second, we use the points estimated from the SDP solution as, the initial iterate for a gradient-descent method to further refine the estimated points. A lower bound obtained from the optimal SDP objective value can be used to check the solution quality. Experimental results are presented to, validate our methods and show that they outperform existing SDP methods.
Inspired by a question of Lovasz, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal called theta bodies of the ideal. These relaxations g...
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Inspired by a question of Lovasz, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal called theta bodies of the ideal. These relaxations generalize Lovasz's construction of the theta body of a graph. We establish a relationship between theta bodies and Lasserre's relaxations for real varieties which allows, in many cases, for theta bodies to be expressed as feasible regions of semidefinite programs. Examples from combinatorial optimization are given. Lovasz asked to characterize ideals for which the first theta body equals the closure of the convex hull of its real variety. We answer this question for vanishing ideals of finite point sets via several equivalent characterizations. We also give a geometric description of the first theta body for all ideals.
This paper studies the representation of a positive polynomial f on a closed semialgebraic set S := {x is an element of R-n vertical bar g(i)(x) = 0, i = 1, ... , l, h(j)(x) >= 0, j = 1, ... , m} modulo the so-call...
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This paper studies the representation of a positive polynomial f on a closed semialgebraic set S := {x is an element of R-n vertical bar g(i)(x) = 0, i = 1, ... , l, h(j)(x) >= 0, j = 1, ... , m} modulo the so-called critical ideal I(f, S) of f on S. Under a constraint qualification condition, it is demonstrated that, if either f > 0 on S or f >= 0 on S and the critical ideal I(f, S) is radical, then f belongs to the preordering generated by the polynomials h(1), ... , h(m) modulo the critical ideal I(f, S). These facts imply that we can find a natural sequence of semidefinite programs whose optimal values converge monotonically, increasing to the infimum value f* := inf(x is an element of S) f(x) of f on S, provided that the infimum value is attained at some point. Besides, we shall construct a finite set in R containing the infimum value f*. Moreover, some relations between the Fedoryuk [Soviet Math. Dokl., 17 (1976), pp. 486-490] and Malgrange [Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, Lecture Notes in Phys. 126, Springer, Berlin, 1980, pp. 170-177] conditions and coercivity for polynomials, which are bounded from below on S, are also established. In particular, a sufficient condition for f to attain its infimum on S is derived from these facts. We also show that every polynomial f, which is bounded from below on S, can be approximated in the l(1)-norm of coefficients by a sequence of polynomials f(epsilon) that are coercive. Finally, it is shown that almost every linear polynomial function, which is bounded from below on S, attains its infimum on S and has the same asymptotic growth at infinity.
We consider optimization problems with polynomial inequality constraints in noncommuting variables. These noncommuting variables are viewed as bounded operators on a Hilbert space whose dimension is not fixed and the ...
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We consider optimization problems with polynomial inequality constraints in noncommuting variables. These noncommuting variables are viewed as bounded operators on a Hilbert space whose dimension is not fixed and the associated polynomial inequalities as semidefinite positivity constraints. Such problems arise naturally in quantum theory and quantum information science. To solve them, we introduce a hierarchy of semidefinite programming relaxations which generates a monotone sequence of lower bounds that converges to the optimal solution. We also introduce a criterion to detect whether the global optimum is reached at a given relaxation step and show how to extract a global optimizer from the solution of the corresponding semidefinite programming problem.
Linear programming, LP, problems with finite optimal value have a zero duality gap and a primal-dual strictly complementary optimal solution pair. On the other hand, there exist semidefinite programming, SDP, problems...
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Linear programming, LP, problems with finite optimal value have a zero duality gap and a primal-dual strictly complementary optimal solution pair. On the other hand, there exist semidefinite programming, SDP, problems which have a nonzero duality gap (different primal and dual optimal values;not both infinite). The duality gap is assured to be zero if a constraint qualification, e.g., Slater's condition (strict feasibility) holds. Measures of strict feasibility, also called distance to infeasibility, have been used in complexity analysis, and, it is known that (near) loss of strict feasibility results in numerical difficulties. In addition, there exist SDP problems which have a zero duality gap but no strict complementary primal-dual optimal solution. We refer to these problems as hard instances of SDP. The assumption of strict complementarity is essential for asymptotic superlinear and quadratic rate convergence proofs. In this paper, we introduce a procedure for generating hard instances of SDP with a specified complementarity nullity (the dimension of the common nullspace of primal-dual optimal pairs). We then show, empirically, that the complementarity nullity correlates well with the observed local convergence rate and the number of iterations required to obtain optimal solutions to a specified accuracy, i.e., we show, even when Slater's condition holds, that the loss of strict complementarity results in numerical difficulties. We include two new measures of hardness that correlate well with the complementarity nullity.
In this article, a primal-dual interior-point algorithm for semidefinite programming that can be used for analysing e.g. polytopic linear differential inclusions is tailored in order to be more computationally efficie...
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In this article, a primal-dual interior-point algorithm for semidefinite programming that can be used for analysing e.g. polytopic linear differential inclusions is tailored in order to be more computationally efficient. The key to the speedup is to allow for inexact search directions in the interior-point algorithm. These are obtained by aborting an iterative solver for computing the search directions prior to convergence. A convergence proof for the algorithm is given. Two different preconditioners for the iterative solver arc proposed. The speedup is in many cases more than an order of magnitude. Moreover, the proposed algorithm can be used to analyse much larger problems as compared to what is possible with off-the-shelf interior-point solvers.
We characterize the spectral behavior of a primal Schur-complement-based block diagonal preconditioner for saddle point systems, subject to low-rank modifications. This is motivated by a desire to reduce as much as po...
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We characterize the spectral behavior of a primal Schur-complement-based block diagonal preconditioner for saddle point systems, subject to low-rank modifications. This is motivated by a desire to reduce as much as possible the computational cost of matrix-vector products with the (1,1) block, while keeping the eigenvalues of the preconditioned matrix reasonably clustered. The formulation leads to a perturbed hyperbolic quadratic eigenvalue problem. We derive interlacing results, highlighting the differences between this problem and perturbed linear eigenvalue problems. As an example, we consider primal-dual interior point methods for semidefinite programs, and express the eigenvalues of the preconditioned matrix in terms of the centering parameter.
Digital signal processing requires digital filters with variable frequency characteristics. A variable digital filter (VDF) is a filter whose frequency characteristics can be easily and instantaneously changed. In thi...
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Digital signal processing requires digital filters with variable frequency characteristics. A variable digital filter (VDF) is a filter whose frequency characteristics can be easily and instantaneously changed. In this paper, we present a design method for variable linear-phase finite impulse response (FIR) filters with multiple variable factors and a reduction method for the number of polynomial coefficients. The obtained filter has a high piecewise attenuation in the stopband. The stopband edge and the position and magnitude of the high piecewise stopband attenuation can be varied by changing some parameters. Variable parameters are normalized in this paper. An optimization methodology known as semidefinite programming (SDP) is used to design the filter. In addition, we present that the proposed VDF can be implemented using the Farrow structure, which. suitable for real time signal processing. The usefulness of the proposed filter is demonstrated through examples.
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