This paper presents a smoothing heuristic for an NP-hard combinatorial problem. Starting with a convex Lagrangian relaxation, a pathfollowing method is applied to obtain good solutions while gradually transforming the...
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This paper presents a smoothing heuristic for an NP-hard combinatorial problem. Starting with a convex Lagrangian relaxation, a pathfollowing method is applied to obtain good solutions while gradually transforming the relaxed problem into the original problem formulated with an exact penalty function. Starting points are drawn using different sampling techniques that use randomization and eigenvectors. The dual point that defines the convex relaxation is computed via eigenvalue optimization using subgradient techniques. The proposed method turns out to be competitive with the most recent ones. The idea presented here is generic and can be generalized to all box-constrained problems where convex Lagrangian relaxation can be applied. Furthermore, to the best of our knowledge, this is the first time that a Lagrangian heuristic is combined with pathfollowing techniques.
Many theoretical and algorithmic results in semidefinite programming are based on the assumption that Slater's constraint qualification is satisfied for the primal and the associated dual problem. We consider semi...
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Many theoretical and algorithmic results in semidefinite programming are based on the assumption that Slater's constraint qualification is satisfied for the primal and the associated dual problem. We consider semidefinite problems with zero duality gap for which Slater's condition fails for at least one of the primal and dual problem. We propose a numerically reasonable way of dealing with such semidefinite programs. The new method is based on a standard search direction with damped Newton steps towards primal and dual feasibility.
We consider the design of approximation algorithms for a number of maximum graph partitioning problems, among others MAX-k-CUT, MAX-k-DENSE-SUBGRAPH, and MAX-k-DIRECTED-UNCUT. We present a new version of the semidefni...
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We consider the design of approximation algorithms for a number of maximum graph partitioning problems, among others MAX-k-CUT, MAX-k-DENSE-SUBGRAPH, and MAX-k-DIRECTED-UNCUT. We present a new version of the semidefnite relaxation scheme along with a better analysis, extending work of Halperin and Zwick (2002). This leads to an improvement over known approximation factors for such problems. The key to the improvement is the following new technique: It was already observed by Han et al. (2002) that a parameter-driven choice of the random hyperplane can lead to better approximation factors than obtained by Goemans and Williamson (1995). But it remained difficult to find a "good" set of parameters. In this paper, we analyze random hyperplanes depending on several new parameters. We prove that a sub-optimal choice of these parameters can be obtained by the solution of a linear program which leads to the desired improvement of the approximation factors. In this fashion a more systematic analysis of the semidefinite relaxation scheme is obtained.
This paper attempts to demonstrate that a modern optimization methodology known as semidefinite programming (SDP) can be served as the algorithmic core of a unified design tool for a variety of two-dimensional (2-D) d...
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This paper attempts to demonstrate that a modern optimization methodology known as semidefinite programming (SDP) can be served as the algorithmic core of a unified design tool for a variety of two-dimensional (2-D) digital filters. Representative SDP-based designs presented in the paper include minimax and weighted least-squares designs of FIR filters with continuous and discrete coefficients, and minimax design of stable separable-denominator HR filters. Our studies are motivated by the fact that SDP as a subclass of convex programming can be solved efficiently using recently developed interior-point methods and, more importantly, constraints on amplitude/phase responses in certain frequency regions and on stability (for UR filters), that are often encountered in many filter design problems, can be formulated in a natural way as linear matrix inequalities (LMI) which allow SDP to apply. Design examples for each class of filters are included to demonstrate that SDP-based methods can in many cases be useful in producing optimal or near-optimal 2-D filters with reduced computational complexity.
Given an undirected graph G = (V, E) with \V\ = n and an integer k between 0 and n, the maximization graph partition (MAX-GP) problem is to determine a subset S C V of k nodes such that an objective function w(S) is m...
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Given an undirected graph G = (V, E) with \V\ = n and an integer k between 0 and n, the maximization graph partition (MAX-GP) problem is to determine a subset S C V of k nodes such that an objective function w(S) is maximized. The MAX-GP problem can be formulated as a binary quadratic program and it is NP-hard. semidefinite programming (SDP) relaxations Of Such quadratic programs have been used to design approximation algorithms with guaranteed performance ratios for various MAX-GP problems. Based on several earlier results, we present an improved rounding method using an SDP relaxation, and establish improved approximation ratios for several MAX-GP problems, including Dense-Subgraph, Max-Cut, Max-Not-Cut, and Max-Vertex-Cover.
In each iteration of an interior-point method for semidefinite programming, the maximum step-length that can be taken by the iterate while maintaining the positive semidefiniteness constraint needs to be estimated. In...
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In each iteration of an interior-point method for semidefinite programming, the maximum step-length that can be taken by the iterate while maintaining the positive semidefiniteness constraint needs to be estimated. In this note, we show how the maximum step-length can be estimated via the Lanczos iteration, a standard iterative method for estimating the extremal eigenvalues of a matrix. We also give a posteriori error bounds for the estimate. Numerical results on the performance of the proposed method against two commonly used methods for calculating step-lengths (backtracking via Cholesky factorizations and exact eigenvalues computations) are included.
In a recent work, Boyd, Diaconis and Xiao introduced a semidefinite programming approach for computing the fastest mixing Markov chain on a graph of allowed transitions, given a target stationary distribution. In this...
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In a recent work, Boyd, Diaconis and Xiao introduced a semidefinite programming approach for computing the fastest mixing Markov chain on a graph of allowed transitions, given a target stationary distribution. In this paper, we show that standard mixing time analysis techniques-variational characterizations, conductance, canonical paths-can be used to give simple, nontrivial lower and upper bounds on the fastest mixing time. To test the applicability of this idea, we consider several detailed examples including the Glauber dynamics of the Ising model.
This paper describes a load dispatch method which minimizes power cost-[fuel cost]/[electric output]-for a power system with thermal plants and energy storage facilities. The proposed method employs fractional program...
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This paper describes a load dispatch method which minimizes power cost-[fuel cost]/[electric output]-for a power system with thermal plants and energy storage facilities. The proposed method employs fractional programming to convert a minimization problem with fractional objective function to a series of quadratic minimization problem, and semidefinite programming to solve converted problems. The method provides the optimum time-dependent power output/input and storage level of energy storage facilities as well as time-dependent power output of thermal plants. The method has been applied to a power system with five thermal plants, two energy storage facilities of various performances, and five load demands. The optimum load scheme of four time mesh points is obtained for the thermal plants and energy storage facilities. The fractional programming successfully converges the optimal scheme through a few iterations. The semidefinite programming deals with a variable matrix of 164 dimensions, and 185 inequality constraints. (C) 2001 Scripta Technica.
semidefinite programming based approximation algorithms, such as the Goemans and Williamson approximation algorithm for the MAX CUT problem, are usually shown to have certain performance guarantees using local ratio t...
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semidefinite programming based approximation algorithms, such as the Goemans and Williamson approximation algorithm for the MAX CUT problem, are usually shown to have certain performance guarantees using local ratio techniques. Are the bounds obtained in this way tight? This problem was considered before by Karloff [SIAM J, Comput., 29 (1999), pp. 336-350] and by Alon and Sudakov [Combin. Probab. Comput., 9 (2000), pp. 1-12]. Here we further extend their results and show, for the first time, that the local analyses of the Goemans and Williamson MAX CUT algorithm, as well as its extension by Zwick, are tight for every possible relative size of the maximum cut in the sense that the expected value of the solutions obtained by the algorithms may be as small as the analyses ensure. We also obtain similar results for a related problem. Our approach is quite general and could possibly be applied to some additional problems and algorithms.
In this paper we find a characterization for when a multivariable trigonometric polynomial can be written as a sum of squares. In addition, the truncated moment problem is addressed. A numerical algorithm for finding ...
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In this paper we find a characterization for when a multivariable trigonometric polynomial can be written as a sum of squares. In addition, the truncated moment problem is addressed. A numerical algorithm for finding a sum of squares representation is presented as well. In the one-variable case, the algorithm finds a spectral factorization. The latter may also be used to find inner-outer factorizations.
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