Book ChapterDOI
The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices
Jane Cullum,W. E. Donath,P. Wolfe +2 more
- pp 35-55
TLDR
In this paper, the sum of the q algebraically largest eigenvalues of any real symmetric matrix as a function of the diagonal entries of the matrix is derived and a convergent procedure is presented for determining a minimizing point of any such sum subject to the condition that the trace of the original matrix is held constant.Abstract:
Properties of the sum of the q algebraically largest eigenvalues of any real symmetric matrix as a function of the diagonal entries of the matrix are derived Such a sum is convex but not necessarily everywhere differentiable A convergent procedure is presented for determining a minimizing point of any such sum subject to the condition that the trace of the matrix is held constant An implementation of this procedure is described and numerical results are includedread more
Citations
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Book
Linear Matrix Inequalities in System and Control Theory
TL;DR: In this paper, the authors present a brief history of LMIs in control theory and discuss some of the standard problems involved in LMIs, such as linear matrix inequalities, linear differential inequalities, and matrix problems with analytic solutions.
Journal ArticleDOI
Semidefinite programming
Lieven Vandenberghe,Stephen Boyd +1 more
TL;DR: A survey of the theory and applications of semidefinite programs and an introduction to primaldual interior-point methods for their solution are given.
Book
Algorithms for VLSI Physical Design Automation
TL;DR: This book is a core reference for graduate students and CAD professionals and presents a balance of theory and practice in a intuitive manner.
Book
Fastest mixing Markov chain on a graph
TL;DR: The Lagrange dual of the fastest mixing Markov chain problem is derived, which gives a sophisticated method for obtaining (arbitrarily good) bounds on the optimal mixing rate, as well as the optimality conditions.
Journal ArticleDOI
An Interior-Point Method for Semidefinite Programming
TL;DR: A new interior-point-based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices is proposed.
References
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Book
The algebraic eigenvalue problem
TL;DR: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography.
Journal ArticleDOI
Validation of subgradient optimization
TL;DR: It is concluded that the “relaxation” procedure for approximately solving a large linear programming problem related to the traveling-salesman problem shows promise for large-scale linear programming.
Journal ArticleDOI
Lower bounds for the partitioning of graphs
W. E. Donath,Alan J. Hoffman +1 more
TL;DR: In this paper, it was shown that the right-hand side is a concave function of the diagonal matrix U such that the sum of the adjacency matrix of the graph plus all the elements of the sum matrix is zero.
Book ChapterDOI
A method of conjugate subgradients for minimizing nondifferentiable functions
TL;DR: In this paper, an algorithm for finding the minimum of any convex, not necessarily differentiable, function f of several variables is described, which yields a sequence of points tending to the solution of the problem, if any, requiring only the calculation of f and one subgradient of f at designated points.