A standard quadratic optimization problem (StQP) consists in minimizing a quadratic form over a simplex. Among the problems which can be transformed into a StQP are the general quadratic problem over a polytope, and t...
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A standard quadratic optimization problem (StQP) consists in minimizing a quadratic form over a simplex. Among the problems which can be transformed into a StQP are the general quadratic problem over a polytope, and the maximum clique problem in a graph. In this paper we present several new polynomial-time bounds for StQP ranging from very simple and cheap ones to more complex and tight constructions. The main tools employed in the conception and analysis of most bounds are semidefinite programming and decomposition of the objective function into a sum of two quadratic functions, each of which is easy to minimize. We provide a complete diagram of the dominance, incomparability, or equivalence relations among the bounds proposed in this and in previous works. In particular, we show that one of our new bounds dominates all the others. Furthermore, a specialization of such bound dominates Schrijver's improvement of Lovasz's theta function bound for the maximum size of a clique in a graph.
We design a randomized polynomial time algorithm which, given a 3-tensor of real numbers A = {a(ijk)}(i,j,k=1)(n) such that for all i, j, k is an element of {1,..., n} we have a(ijk) = a(ikj) = a(kji) = a(jik) = a(kij...
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We design a randomized polynomial time algorithm which, given a 3-tensor of real numbers A = {a(ijk)}(i,j,k=1)(n) such that for all i, j, k is an element of {1,..., n} we have a(ijk) = a(ikj) = a(kji) = a(jik) = a(kij) = a(jki) and a(iik) = a(ijj) = a(iji) = 0, computes a number Alg(A) which satisfies with probability at least 1/2, Omega(root log n/n t) center dot max(x) is an element of{-1,1}(n) Sigma(n)(i, j, k= 1) a(ijk)x(i)x(j)x(k) <= Alg(A) <= max(x is an element of{-1,1})(n) Sigma(n)(i, j, k=1) a(ijk)x(i)x(j)x(k). On the other hand, we show via a simple reduction from a result of H astad and Venkatesh [ Random Structures Algorithms, 25 ( 2004), pp. 117 - 149] that under the assumption NP not subset of. DTIME(n((log n)O(1))), for every is an element of > 0 there is no algorithm that approximates max(x is an element of{-1,1})(n) Sigma(n)(i, j, k=1) a(ijk)x(i)x(j)x(k) within a factor of 2(log n)(1-is an element of) in time 2(log n)(O(1)). Our algorithm is based on a reduction to the problem of computing the diameter of a convex body in R-n with respect to the L-1 norm. We show that it is possible to do so up to a multiplicative error of O(root n/log n), while no randomized polynomial time algorithm can achieve accuracy o(root n/log n). This resolves a question posed by Brieden et al. in [Mathematika, 48 (2001), pp. 63 - 105]. We apply our new algorithm to improve the algorithm of H astad and Venkatesh for the Max-E3-Lin-2 problem. Given an overdetermined system epsilon of N linear equations modulo 2 in n <= N Boolean variables such that in each equation only three distinct variables appear, the goal is to approximate in polynomial time the maximum number of satisfiable equations in epsilon minus N/2 (i.e., we subtract the expected number of satisfied equations in a random assignment). H astad and Venkatesh obtained an algorithm which approximates this value up to a factor of O(root N). We obtain an O(root n/log n) approximation algorithm. By relating this problem
We propose a method to calculate lower and upper bounds of some exponential multivariate integrals using moment relaxations and show that they asymptotically converge to the value of the integrals when the moment degr...
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We propose a method to calculate lower and upper bounds of some exponential multivariate integrals using moment relaxations and show that they asymptotically converge to the value of the integrals when the moment degree increases. We report computational results for integrals involving the normal distribution and exponential order statistic probabilities. (C) 2007 Elsevier B.V. All rights reserved.
We revisit a regularization technique of Meszaros for handling free variables within interior-point methods for conic linear optimization. We propose a simple computational strategy, supported by a global convergence ...
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We revisit a regularization technique of Meszaros for handling free variables within interior-point methods for conic linear optimization. We propose a simple computational strategy, supported by a global convergence analysis, for handling the regularization. Using test problems from benchmark suites and recent applications, we demonstrate that the modern code SDPT3 modified to incorporate the proposed regularization is able to achieve the same or significantly better accuracy over standard options of splitting variables, using a quadratic cone, and solving indefinite systems.
A new algorithm is proposed for generating scenarios from a partially specified symmetric multivariate distribution. The algorithm generates samples which match the first two moments exactly, and match the marginal fo...
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A new algorithm is proposed for generating scenarios from a partially specified symmetric multivariate distribution. The algorithm generates samples which match the first two moments exactly, and match the marginal fourth moments approximately, using a semidefinite programming procedure. The performance of the algorithm is illustrated by a numerical example. (C) 2008 Elsevier B.V. All rights reserved.
We show that all the coefficients of the polynomial are nonnegative whenever <= 13 is a nonnegative integer and A and B are positive semidefinite matrices of the same size. This has previously been known only for &...
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We show that all the coefficients of the polynomial are nonnegative whenever <= 13 is a nonnegative integer and A and B are positive semidefinite matrices of the same size. This has previously been known only for <= 7. The validity of the statement for arbitrary m has recently been shown to be equivalent to the Bessis-Moussa-Villani conjecture from theoretical physics. In our proof, we establish a connection to sums of hermitian squares of polynomials in noncommuting variables and to semidefinite programming. As a by-product we obtain an example of a real polynomial in two noncommuting variables having nonnegative trace on all symmetric matrices of the same size, yet not being a sum of hermitian squares and commutators.
This paper discusses a method for estimating noise covariances from process data. In linear stochastic state-space representations the true noise covariances are generally unknown in practical applications. Using esti...
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This paper discusses a method for estimating noise covariances from process data. In linear stochastic state-space representations the true noise covariances are generally unknown in practical applications. Using estimated covariances a Kalman filter can be tuned in order to increase the accuracy of the state estimates. There is a linear relationship between covariances and autocovariance. Therefore, the covariance estimation problem can be stated as a least-squares problem, which can be solved as a symmetric semidefinite least-squares problem. This problem is convex and can be solved efficiently by interior-point methods. A numerical algorithm for solving the symmetric is able to handle systems with mutually correlated process noise and measurement noise. (c) 2007 Elsevier Ltd. All rights reserved.
We study various semidefinite programming (SDP) formulations for Vertex Cover by adding different constraints to the standard formulation. We show that Vertex Cover cannot be approximated better than 2 - O(root log lo...
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We study various semidefinite programming (SDP) formulations for Vertex Cover by adding different constraints to the standard formulation. We show that Vertex Cover cannot be approximated better than 2 - O(root log log n/log n) even when we add the so-called pentagonal inequality constraints to the standard SDP formulation, and thus almost meet the best upper bound known due to Karakostas [Proceedings of the 32nd International Colloquium on Automata, Languages and programming, 2005], of 2 - Omega(root 1/log n). We further show the surprising fact that by strengthening the SDP with the (intractable) requirement that the metric interpretation of the solution embeds into l(1) with no distortion, we get an exact relaxation (integrality gap is 1), and on the other hand, if the solution is arbitrarily close to being l(1) embeddable, the integrality gap is 2-o(1). Finally, inspired by the above findings, we use ideas from the integrality gap construction of Charikar [SODA '02: Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2002, pp. 616-620] to provide a family of simple examples for negative type metrics that cannot be embedded into l(1) with distortion better than 8/7 - is an element of. To this end we prove a new isoperimetric inequality for the hypercube.
The goal of this paper is to study the so-called worst-case or robust optimal design problem for minimal compliance. In the context of linear elasticity we seek an optimal shape which minimizes the largest, or worst, ...
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The goal of this paper is to study the so-called worst-case or robust optimal design problem for minimal compliance. In the context of linear elasticity we seek an optimal shape which minimizes the largest, or worst, compliance when the loads are subject to some unknown perturbations. We first prove that, for a fixed shape, there exists indeed a worst perturbation ( possibly non unique) that we characterize as the maximizer of a nonlinear energy. We also propose a stable algorithm to compute it. Then, in the framework of Hadamard method, we compute the directional shape derivative of this criterion which is used in a numerical algorithm, based on the level set method, to find optimal shapes that minimize the worst-case compliance. Since this criterion is usually merely directionally differentiable, we introduce a semidefinite programming approach to select the best descent direction at each step of a gradient method. Numerical examples are given in 2-d and 3-d.
First, three different but related output regulation performance criteria for the linear time-varying system are defined in the discrete-time domain, namely, the peak impulse response, peak output variance, and averag...
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First, three different but related output regulation performance criteria for the linear time-varying system are defined in the discrete-time domain, namely, the peak impulse response, peak output variance, and average output variance per unit time. Then they are extended for switched linear systems and Markovian jump linear systems, and characterized by an increasing union of finite-dimensional linear matrix inequality conditions. Finally, the infinite-horizon suboptimal LQG control problem, which aims to maintain the average output variance below a given level subject to the uniform exponential stability of the closed-loop system, is solved for both switched linear systems and Markovian jump linear systems;the solution is given by a dynamic linear output feedback controller that not only perfectly observes the present mode but also recalls a finite number of past modes.
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