This paper presents a feasible primal algorithm for linear semidefinite programming. The algorithm starts with a strictly feasible solution, but in case where no such a solution is known, an application of the algorit...
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This paper presents a feasible primal algorithm for linear semidefinite programming. The algorithm starts with a strictly feasible solution, but in case where no such a solution is known, an application of the algorithm to an associate problem allows to obtain one. Finally, we present some numerical experiments which show that the algorithm works properly.
We consider semidefinite programming problems on which a permutation group is acting. We describe a general technique to reduce the size of such problems, exploiting the symmetry. The technique is based on a low-order...
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We consider semidefinite programming problems on which a permutation group is acting. We describe a general technique to reduce the size of such problems, exploiting the symmetry. The technique is based on a low-order matrix *-representation of the commutant (centralizer ring) of the matrix algebra generated by the permutation matrices. We apply it to extending a method of de Klerk et al. that gives a semidefinite programming lower bound to the crossing number of complete bipartite graphs. It implies that cr(K-8,K-n ) >= 2.9299n(2)-6n, cr(K-9,K-n ) >= 3.8676n(2)-8n, and (for any m >= 9) [GRAPHICS] where Z(m,n) is the Zarankiewicz number [1/4(m -1)(2)][1/4(n -1)(2)], which is the conjectured value of cr(K (m,n) ). Here the best factor previously known was 0.8303 instead of 0.8594.
Convex-optimization techniques are very popular in the very large-scale-integration design society due to their guaranteed convergence to a global optimal point. The table data need to be fitted into convex forms to b...
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Convex-optimization techniques are very popular in the very large-scale-integration design society due to their guaranteed convergence to a global optimal point. The table data need to be fitted into convex forms to be used in the convex optimization problems. Fitting the tables into posynomials, which are analytically convex under logarithmic transformation, may suffer from the excessive fitting errors as the fitting problem is nonconvex. In this paper, we propose to directly adjust the lookup-table values into a numerically convex lookup table without any explicit analytical form. We show that numerically "convexifying" the lookup-table data with minimum perturbation can be formulated as a convex semidefinite optimization problem, and hence, optimality can be reached in polynomial time. We also propose three algorithms to make the table data smooth to enable faster convergence of the convex optimizer. Results from extensive experiments on industrial cell libraries demonstrate 9.6 x improvement in fitting error over a well-developed posynomial-fitting procedure. We illustrate the effectiveness of this model in a convex optimization problem by providing results for using our model in the optimal gate sizing of standard cells. We observe a 5.07% improvement in the delay of International Symposium on Circuits and Systems (ISCAS) benchmark circuits over the posynomial-fitting procedure.
Korkin and Zolotarev showed that if Sigma(i) A(i)(x(i) - Sigma(j > i) alpha(ij)x(j))(2) is the Lagrange expansion of a Korkin-Zolotarev (KZ-) reduced positive definite quadratic form, then A(i + 1) >= 3/4 A(i) a...
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Korkin and Zolotarev showed that if Sigma(i) A(i)(x(i) - Sigma(j > i) alpha(ij)x(j))(2) is the Lagrange expansion of a Korkin-Zolotarev (KZ-) reduced positive definite quadratic form, then A(i + 1) >= 3/4 A(i) and A(i) (+) (2) 2/3 A(i). They showed that the implied bound A(5) >= 4/9 A(1) is not attained by any KZ- reduced form. We propose a method to optimize numerically over the set of Lagrange expansions of KZ-reduced quadratic forms using a semidefinite relaxation combined with a branch and bound process. We use a rounding technique to derive exact results from the numerical data. Applying these methods, we prove several new linear inequalities on the A(i) of any KZ- reduced form, one of them being A(i + 4) >= (15/32 - 2.10(-5)) A(i). We also give a form with A(5) = 15/32 A(1). These new inequalities are then used to study the cone of outer coefficients of KZ-reduced forms, to find bounds on Hermite's constant, and to give better estimates on the quality of k-block KZ-reduced lattice bases.
We investigate optimal encoding and retrieval of digital data, when the storage/communication medium is described by quantum mechanics. We assume an m-ary alphabet with arbitrary prior distribution, and an n-dimension...
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We investigate optimal encoding and retrieval of digital data, when the storage/communication medium is described by quantum mechanics. We assume an m-ary alphabet with arbitrary prior distribution, and an n-dimensional quantum system. Under these constraints, we seek an encoding-retrieval setup, comprised of code-states and a quantum measurement, which maximizes the probability of correct detection. In our development, we consider two cases. In the first, the measurement is predefined and we seek the optimal code-states. In the second, optimization is performed on both the code-states and the measurement. We show that one cannot outperform "pseudo-classical transmission," in which we transmit n symbols with orthogonal code-states, and discard the remaining symbols. However, such pseudo-classical transmission is not the only optimum. We fully characterize the collection of optimal setups, and briefly discuss the links between our findings and applications such as quantum key distribution and quantum computing.
A class of important problems in structural mechanics leads to optimization problems with linear objective functions and constraints consisting in (a) linear equalities and (b) inequalities imposed by the material str...
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A class of important problems in structural mechanics leads to optimization problems with linear objective functions and constraints consisting in (a) linear equalities and (b) inequalities imposed by the material strength, the so-called failure criteria. It is shown that a wide variety of failure criteria can be represented as systems of either second-order cone or semidefinite constraints, giving rise to respective optimization problems.
We propose a Positivstellensatz for trigonometric polynomials that is simpler than its correspondent for polynomials of real variable. Using it, we give a stability test for multidimensional systems, consisting of che...
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We propose a Positivstellensatz for trigonometric polynomials that is simpler than its correspondent for polynomials of real variable. Using it, we give a stability test for multidimensional systems, consisting of checking the feasibility of a linear matrix inequality. The same result is used for a robust stability test for systems whose coefficients depend polynomially on some bounded parameters. The new tests are either more accurate or faster than previous ones.
We consider the problem of computing the outer-radii of point sets. In this problem, we are given integers n, d, and k, where k 0 when d - k is a fixed constant [M. Badoiu, S. Har-Peled, and P. Indyk, in Proceedings ...
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We consider the problem of computing the outer-radii of point sets. In this problem, we are given integers n, d, and k, where k <= d, and a set P of n points in R-d. The goal is to compute the outer k-radius of P, denoted by R-k(P), which is the minimum over all (d - k)-dimensional flats F of max(p epsilon P) d(p, F), where d(p, F) is the Euclidean distance between the point p and flat F. Computing the radii of point sets is a fundamental problem in computational convexity with many significant applications. The problem admits a polynomial time algorithm when the dimension d is constant [U. Faigle, W. Kern, and M. Streng, Math. Program., 73 (1996), pp. 1-5]. Here we are interested in the general case in which the dimension d is not fixed and can be as large as n, where the problem becomes NP-hard even for k = 1. It is known that R-k(P) can be approximated in polynomial time by a factor of (1 + epsilon) for any epsilon > 0 when d - k is a fixed constant [M. Badoiu, S. Har-Peled, and P. Indyk, in Proceedings of the ACM Symposium on the Theory of Computing, 2002;S. Har-Peled and K. Varadarajan, in Proceedings of the ACM Symposium on Computing Geometry, 2002]. A polynomial time algorithm that guarantees a factor of O(root log n) approximation for R-1(P), the width of the point set P, is implied by the results of Nemirovski, Roos, and Terlaky [Math. Program., 86 (1999), pp. 463-473] and Nesterov [Handbook of semidefinite programming Theory, Algorithms, Kluwer Academic Publishers, Norwell, MA, 2000]. In this paper, we show that R-k(P) can be approximated by a ratio of O(root log n) for any 1 <= k <= d, thus matching the previously best known ratio for approximating the special case R-1(P), the width of point set P. Our algorithm is based on semidefinite programming relaxation with a new mixed deterministic and randomized rounding procedure. We also prove an inapproximability result that gives evidence that our approximation algorithm is doing well for a large range of k.
The binary quadratic knapsack problem maximizes a quadratic objective function subject to a linear capacity constraint. Due to its simple structure and challenging difficulty it has been studied intensively during the...
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The binary quadratic knapsack problem maximizes a quadratic objective function subject to a linear capacity constraint. Due to its simple structure and challenging difficulty it has been studied intensively during the last two decades. The present paper gives a survey of upper bounds presented in the literature, and show the relative tightness of several of the bounds. Techniques for deriving the bounds include relaxation from upper planes, linearization, reformulation, Lagrangian relaxation, Lagrangian decomposition, and semidefinite programming. A short overview of heuristics, reduction techniques, branch-and-bound algorithms and approximation results is given, followed by an overview of valid inequalities for the quadratic knapsack polytope. The paper is concluded by an experimental study where the upper bounds presented are compared with respect to strength and computational effort. (c) 2006 Elsevier B.V. All rights reserved.
We consider the problem of estimating a vector x in the linear model Ax approximate to y, where A is a block circulant ( BC) matrix with N blocks and x is assumed to have a weighted norm bound. In the case where both ...
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We consider the problem of estimating a vector x in the linear model Ax approximate to y, where A is a block circulant ( BC) matrix with N blocks and x is assumed to have a weighted norm bound. In the case where both A and y are subjected to noise, we propose a minimax mean-squared error (MSE) approach in which we seek the linear estimator that minimizes the worst-case MSE over a BC structured uncertainty region. For an arbitrary choice of weighting, we show that the minimax MSE estimator can be formulated as a solution to a semidefinite programming problem ( SDP), which can be solved efficiently. For a Euclidean norm bound on x, the SDP is reduced to a simple convex program with N + 1 unknowns. Finally, we demonstrate through an image deblurring example the potential of the minimax MSE approach in comparison with other conventional methods.
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