We construct new approximation algorithms for MAX SET SPLITTING and MAX NOT-ALL-EQUAL SAT which when combined with existing algorithms give the best approximation results so far for these problems. Furthermore, we sol...
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We construct new approximation algorithms for MAX SET SPLITTING and MAX NOT-ALL-EQUAL SAT which when combined with existing algorithms give the best approximation results so far for these problems. Furthermore, we solve a linear program to find an upper bound on the performance ratio. This linear program can also be used to see which of the contributing algorithms it is possible to exclude from the combined algorithm without affecting its performance ratio. (C) 1998 Elsevier Science B.V.
In this paper, we study the spectral methods for graph bisection problems and conclude certain numerical results. Spectral methods contain two topics: eigenvalue bounds for graph bisection width and bisection algorith...
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In this paper, we study the spectral methods for graph bisection problems and conclude certain numerical results. Spectral methods contain two topics: eigenvalue bounds for graph bisection width and bisection algorithms with eigenvector space. Two graph bisection-problems are mainly investigated: finding the best bisection for the equal-size graph bisection problem and choosing the best bisection among a set of graph bisection problems of specified sizes. To compute the optimal eigenvalue bounds, one needs to solve the eigenvalue optimization problem which minimizes the largest eigenvalue of an affine symmetric matrix function. The eigenvalue optimization problems are convex but usually nonsmooth. Thus, we transform the eigenvalue optimization problems into the semidefinite programming problems. Then we apply primal-dual interior point methods to solve the semidefinite programming problems. Finally, we use an eigenvector corresponding to the largest eigenvalue, which has attained the optimal bound, to bisect the graph into two pieces of specified size. The running time of our algorithm for finding the best bisection among a set of graph bisection problems of specified sizes is O(n(3)). (C) 1998 Published by Elsevier Science Ltd. All rights reserved.
This paper presents an overview of some recent, and significant, progress in the theory of optimization problems with perturbations. We put the emphasis on methods based on upper and lower estimates of the objective f...
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This paper presents an overview of some recent, and significant, progress in the theory of optimization problems with perturbations. We put the emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems. These methods allow one to compute expansions of the optimal value function and approximate optimal solutions in situations where the set of Lagrange multipliers is not a singleton, may be unbounded, or is even empty. We give rather complete results for nonlinear programming problems and describe some extensions of the method to more general problems. We illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal.
The theory of the convergence of Krylov subspace iterations for linear systems of equations (conjugate gradients, biconjugate gradients, GMRES, QMR, Bi-CGSTAB, and so on) is reviewed. For a computation of this kind, a...
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The theory of the convergence of Krylov subspace iterations for linear systems of equations (conjugate gradients, biconjugate gradients, GMRES, QMR, Bi-CGSTAB, and so on) is reviewed. For a computation of this kind, an estimated asymptotic convergence factor rho less than or equal to 1 can be derived by solving a problem of potential theory or conformal mapping. Six approximations are involved in relating the actual computation to this scalar estimate. These six approximations are discussed in a systematic way and illustrated by a sequence of examples computed with tools of numerical conformal mapping and semidefinite programming.
The problem of maximizing the determinant of a matrix subject to linear matrix inequalities (LMIs) arises in many fields, including computational geometry, statistics, system identification, experiment design, and inf...
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The problem of maximizing the determinant of a matrix subject to linear matrix inequalities (LMIs) arises in many fields, including computational geometry, statistics, system identification, experiment design, and information and communication theory. It can also be considered as a generalization of the semidefinite programming problem. We give an overview of the applications of the determinant maximization problem, pointing out simple cases where specialized algorithms or analytical solutions are known. We then describe an interior-point method, with a simplified analysis of the worst-case complexity and numerical results that indicate that the method is very efficient, both in theory and in practice. Compared to existing specialized algorithms (where they are available), the interior-point method will generally be slower;the advantage is that it handles a much wider variety of problems.
There are a large number of problems in systems and control that can be formulated as semidefinite programming problems: minimize a linear function subject to linear matrix inequality constraints. In this paper we stu...
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There are a large number of problems in systems and control that can be formulated as semidefinite programming problems: minimize a linear function subject to linear matrix inequality constraints. In this paper we study the effect of perturbations in the semidefinite programming problem on the optimal solution and the optimal value function. A new form of complementarity conditions is proposed and the first-order partial derivatives of the optimal value function with respect to parametric variation are explicitly expressed by the problem data and the optimal solution. Furthermore, the above result is applied to sensitivity analysis of control systems with parametric uncertainties.
An input to the betweenness problem contains m constraints over n real variables (points). Each constraint consists of three points, where one of the points is specified to lie inside the interval defined by the other...
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An input to the betweenness problem contains m constraints over n real variables (points). Each constraint consists of three points, where one of the points is specified to lie inside the interval defined by the other two. The order of the other two points (i.e., which one is the largest and which one is the smallest) is not specified. This problem comes up in questions related to physical mapping in molecular biology. In 1979, Opatrny showed that the problem of deciding whether the n points can be totally ordered while satisfying the m betweenness constraints is NP-complete [SIAM J. Comput., 8 (1979), pp. 111-114]. Furthermore, the problem is MAX SNP complete, and for every alpha > 47/48 finding a total order that satisfies at least alpha of the m constraints is NP-hard (even if all the constraints are satisfiable). It is easy to find an ordering of the points that satisfies 1/3 of them constraints (e.g., by choosing the ordering at random). This paper presents a polynomial time algorithm that either determines that there is no feasible solution or finds a total order that satisfies at least 1/2 of them constraints. The algorithm translates the problem into a set of quadratic inequalities and solves a semidefinite relaxation of them in R-n. The n solution points are then projected on a random line through the origin. The claimed performance guarantee is shown using simple geometric properties of the semidefinite programming (SDP) solution.
In this paper we consider semidefinite programs (SDPs) whose data depend on some unknown but bounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize th...
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In this paper we consider semidefinite programs (SDPs) whose data depend on some unknown but bounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worst-case) objective while satisfying the constraints for every possible value of parameters within the given bounds. Assuming the data matrices are rational functions of the perturbation parameters, we show how to formulate sufficient conditions for a robust solution to exist as SDPs. When the perturbation is "full," our conditions are necessary and sufficient. In this case, we provide sufficient conditions which guarantee that the robust solution is unique and continuous (Holder-stable) with respect to the unperturbed problem's data. The approach can thus be used to regularize ill-conditioned SDPs. We illustrate our results with examples taken from linear programming, maximum norm minimization, polynomial interpolation, and integer programming.
A Chebyshev polynomial of a square matrix A is a monic polynomial p of specified degree that minimizes parallel to p(A)parallel to(2). The study of such polynomials is motivated by the analysis of Krylov subspace iter...
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A Chebyshev polynomial of a square matrix A is a monic polynomial p of specified degree that minimizes parallel to p(A)parallel to(2). The study of such polynomials is motivated by the analysis of Krylov subspace iterations in numerical linear algebra. An algorithm is presented for computing these polynomials based on reduction to a semidefinite program which is then solved by a primal-dual interior point method. Examples of Chebyshev polynomials of matrices are presented, and it is noted that if A is far from normal, the lemniscates of these polynomials tend to approximate pseudospectra of A.
This paper presents some results on quantitative stability of optimal solutions of parameterized optimization problems having nonisolated optima. The analysis is based on the introduced concept of nondegeneracy, which...
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This paper presents some results on quantitative stability of optimal solutions of parameterized optimization problems having nonisolated optima. The analysis is based on the introduced concept of nondegeneracy, which may be of independent interest. Examples of nonlinear and semidefinite programming are discussed.
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