Results 1  10
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19
Determinant maximization with linear matrix inequality constraints
 SIAM Journal on Matrix Analysis and Applications
, 1998
"... constraints ..."
Positive definite completions of partial Hermitian matrices
 Linear Alg. Its Applic
, 1984
"... The question of which partial Hermitian matrices (some entries specified, some free) may be completed to positive definite matrices is addressed. It is shown that if the diagonal entries are specified and principal minors, composed of specified entries, are positive, then, if the undirected graph of ..."
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Cited by 74 (10 self)
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The question of which partial Hermitian matrices (some entries specified, some free) may be completed to positive definite matrices is addressed. It is shown that if the diagonal entries are specified and principal minors, composed of specified entries, are positive, then, if the undirected graph of the specified entries is chordal, a positive definite completion necessarily exists. Furthermore, if this graph is not chordal, then examples exist without positive definite completions. In case a positive definite completion exists, there is a unique matrix, in the class of all positive definite completions, whose determinant is maximal, and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original
An InteriorPoint Method For Approximate Positive Semidefinite Completions
 Comput. Optim. Appl
, 1997
"... Given a nonnegative, symmetric matrix of weights, H, we study the problem of finding an Hermitian, positive semidefinite matrix which is closest to a given Hermitian matrix, A; with respect to the weighting H: This extends the notion of exact matrix completion problems in that, H ij = 0 corresponds ..."
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Cited by 21 (4 self)
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Given a nonnegative, symmetric matrix of weights, H, we study the problem of finding an Hermitian, positive semidefinite matrix which is closest to a given Hermitian matrix, A; with respect to the weighting H: This extends the notion of exact matrix completion problems in that, H ij = 0 corresponds to the element A ij being unspecified (free), while H ij large in absolute value corresponds to the element A ij being approximately specified (fixed). We present optimality conditions, duality theory, and two primaldual interiorpoint algorithms. Because of sparsity considerations, the dualstepfirst algorithm is more efficient for a large number of free elements, while the primalstepfirst algorithm is more efficient for a large number of fixed elements. Included are numerical tests that illustrate the efficiency and robustness of the algorithms. This report (related software) is available by anonymous ftp at orion.uwaterloo.ca in the directory pub/henry/reports (henry/software/compl...
Matrix Completion Problems
 THE ENCYCLOPEDIA OF OPTIMIZATION
, 2001
"... ... In this article we survey some results and provide references about these problems for the following matrix properties: positive semidefinite matrices, Euclidean distance matrices, completely positive matrices, contraction matrices, and matrices of given rank. We treat mainly optimization an ..."
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Cited by 17 (1 self)
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... In this article we survey some results and provide references about these problems for the following matrix properties: positive semidefinite matrices, Euclidean distance matrices, completely positive matrices, contraction matrices, and matrices of given rank. We treat mainly optimization and combinatorial aspects.
Polynomial Instances Of The Positive Semidefinite And Euclidean Distance Matrix Completion Problems
 SIAM J. Matrix Anal. Appl
, 1998
"... Given an undirected graph G = (V; E) with node set V = [1; n], a subset S ` V and a rational vector a 2 Q S[E , the positive semidefinite matrix completion problem consists of determining whether there exists a real symmetric n \Theta n positive semidefinite matrix X = (x ij ) satisfying: x ii = a ..."
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Cited by 13 (5 self)
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Given an undirected graph G = (V; E) with node set V = [1; n], a subset S ` V and a rational vector a 2 Q S[E , the positive semidefinite matrix completion problem consists of determining whether there exists a real symmetric n \Theta n positive semidefinite matrix X = (x ij ) satisfying: x ii = a i (i 2 S) and x ij = a ij (ij 2 E). Similarly, the Euclidean distance matrix completion problem asks for the existence of a Euclidean distance matrix completing a partially defined given matrix. It is not known whether these problems belong to NP. We show here that they can be solved in polynomial time when restricted to the graphs having a fixed minimum fillin; the minimum fillin of graph G being the minimum number of edges needed to be added to G in order to obtain a chordal graph. A simple combinatorial algorithm permits to construct a completion in polynomial time in the chordal case. We also show that the completion problem is polynomially solvable for a class of graphs including wheels of fixed length (assuming all diagonal entries are specified). The running time of our algorithms is polynomially bounded in terms of n and the bitlength of the input a. We also observe that the matrix completion problem can be solved in polynomial time in the real number model for the class of graphs containing no homeomorph of K 4 .
An Efficient Finite Element Method for Submicron IC Capacitance Extraction
 Proc. 26th Design Automation Conference, Las Vegas
, 1989
"... We present an accurate and efficient method for extraction of parasitic capacitances in submicron integrated circuits. The method uses a 3D finite element model in which the conductor charges are approximated by a piecewise linear function on a web of edges located on the surface of the conductors ..."
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Cited by 10 (4 self)
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We present an accurate and efficient method for extraction of parasitic capacitances in submicron integrated circuits. The method uses a 3D finite element model in which the conductor charges are approximated by a piecewise linear function on a web of edges located on the surface of the conductors. This yields a system of Green's function integral equations that is solved by a novel approximate matrix inversion technique that only utilizes the entries corresponding to pairs of finite elements that are physically close to each other. With N representing the size of the layout, this results in time and space complexities of O(N) and O(Ö##N ) respectively. The method has been implemented in an efficient layout to circuit extractor and experimental results are presented. Introduction With the decrease of feature sizes and the increase of chip dimensions in IC technology, the influence of interconnect capacitances on circuit performance is becoming more prominent. Therefore, the need f...
On the Facial Structure of the Set of Correlation Matrices
 SIAM J. ON MATRIX ANALYSIS AND APPLICATIONS
, 1995
"... We study the facial structure of the set E n\Thetan of correlation matrices (i.e., the positive semidefinite matrices with diagonal entries equal to 1). In particular, we determine the possible dimensions for a face, as well as for a polyhedral face of E n\Thetan . It turns out that the spectrum of ..."
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Cited by 8 (2 self)
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We study the facial structure of the set E n\Thetan of correlation matrices (i.e., the positive semidefinite matrices with diagonal entries equal to 1). In particular, we determine the possible dimensions for a face, as well as for a polyhedral face of E n\Thetan . It turns out that the spectrum of face dimensions is lacunary and that E n\Thetan has polyhedral faces of dimension up to ß p 2n. As an application, we describe in detail the faces of E 4\Theta4 . We also discuss results related to optimization over E n\Thetan .
Inertias of Block Band Matrix Completions
"... . This paper classifies the ranks and inertias of hermitian completion for the partially specified 3x3 block band hermitian matrix, (also known as a "bordered matrix") P = 0 B B @ A B ? B C D ? D E 1 C C A : The full set of completion inertias is described in terms of seven linear inequ ..."
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Cited by 5 (0 self)
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. This paper classifies the ranks and inertias of hermitian completion for the partially specified 3x3 block band hermitian matrix, (also known as a "bordered matrix") P = 0 B B @ A B ? B C D ? D E 1 C C A : The full set of completion inertias is described in terms of seven linear inequalities involving inertias and ranks of specified submatrices. The minimal completion rank for P is computed. We study the completion inertias of partially specified hermitian blockband matrices, using a block generalization of the DymGohberg algorithm. At each inductive step, we use our classification of the possible inertias for hermitian completions of bordered matrices. We show that when all the maximal specified submatrices are invertible, any inertia consistent with Poincar'e's inequalities is obtainable. These results generalize the nonblock band results of Dancis ([SIAM J. Matrix Anal. Appl. 14 (1993) pp.813829]). All our results remain valid for real symmetric completions. 2 1...
SEMIDEFINITE AND LAGRANGIAN RELAXATIONS FOR HARD COMBINATORIAL PROBLEMS
"... Semidefinite Programming is currently a very exciting and active area of research. Semidefinite relaxations generally provide very tight bounds for many classes of numerically hard problems. In addition, these relaxations can be solved efficiently by interiorpoint methods. In this ..."
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Cited by 3 (3 self)
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Semidefinite Programming is currently a very exciting and active area of research. Semidefinite relaxations generally provide very tight bounds for many classes of numerically hard problems. In addition, these relaxations can be solved efficiently by interiorpoint methods. In this
Contractive Extension Problems for Matrix Valued Almost Periodic Functions of Several Variables
"... Problems of Nehari type are studied for matrix valued kvariable almost periodic Wiener functions: Find contractive kvariable almost periodic Wiener functions having prespecified Fourier coefficients with indices in a given halfspace of R k . We characterize the existence of a solution, give a co ..."
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Cited by 1 (1 self)
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Problems of Nehari type are studied for matrix valued kvariable almost periodic Wiener functions: Find contractive kvariable almost periodic Wiener functions having prespecified Fourier coefficients with indices in a given halfspace of R k . We characterize the existence of a solution, give a construction of the solution set, and exhibit a particular solution that has a certain maximizing property. These results are used to obtain various distance formulas and multivariable almost periodic extensions of Sarason's theorem. In the periodic case, a generalization of Sarason's theorem is proved using a variation of the commutant lifting theorem. The main results are further applied to a modelmatching problem for multivariable linear filters. Key Words: Almost periodic matrix functions, contractive extensions, Besikovitch space, Hankel operators, Sarason's Theorem, band method, commutant lifting, model matching. AMS Subject Classification: 42A75, 15A54, 47A56, 47A57, 42A82, 47B35. 1 I...