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DETERMINANT MAXIMIZATION WITH LINEAR MATRIX INEQUALITY CONSTRAINTS
"... The problem of maximizing the determinant of a matrix subject to linear matrix inequalities 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 s ..."
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Cited by 229 (18 self)
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The problem of maximizing the determinant of a matrix subject to linear matrix inequalities 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 interiorpoint method, with a simplified analysis of the worstcase 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 interiorpoint method will generally be slower; the advantage is that it handles a much wider variety of problems.
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 128 (9 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
Matrix completion problems
 In The Encyclopedia of Optimization, C.A. Floudas and P.M. Pardalos (eds.), Kluwer, Vol III
, 2001
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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 23 (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...
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 17 (6 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 13 (5 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 13 (3 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 .
1 On the Covariance Completion Problem under a Circulant Structure
"... Abstract — Covariance matrices with a circulant structure arise in the context of discretetime periodic processes and their significance stems also partly from the fact that they can be diagonalized via a Fourier transformation. This note deals with the problem of completion of partially specified ..."
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Abstract — Covariance matrices with a circulant structure arise in the context of discretetime periodic processes and their significance stems also partly from the fact that they can be diagonalized via a Fourier transformation. This note deals with the problem of completion of partially specified circulant covariance matrices. The particular completion that has maximal determinant (i.e., the socalled maximum entropy completion) was considered in Carli etal. [2] where it was shown that if a single band is unspecified and to be completed, the algebraic restriction that enforces the circulant structure is automatically satisfied and that the inverse of the maximizer has a band of zero values that corresponds to the unspecified band in the data—i.e., it has the Dempster property. The purpose of the present note is to develop an independent proof of this result which in fact extends naturally to any number of missing bands as well as arbitrary missing elements. More specifically, we show that this general fact is a direct consequence of the invariance of the determinant under the group of transformations that leave circulant matrices invariant. A description of the complete set of all positive extensions of partially specified circulant matrices is also given and certain connections between such sets and the factorization of certain polynomials in many variables, facilitated by the circulant structure, is highlighted. I.
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 li ..."
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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...
Positive semidefinite matrix completions on chordal graphs and the constraint nondegeneracy in semidefinite programming
, 2008
"... LetG = (V,E) be a graph. In matrix completion theory, it is known that the following two conditions are equivalent: (i) G is a chordal graph; (ii) Every Gpartial positive semidefinite matrix has a positive semidefinite matrix completion. In this paper, we relate these two conditions to constraint n ..."
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LetG = (V,E) be a graph. In matrix completion theory, it is known that the following two conditions are equivalent: (i) G is a chordal graph; (ii) Every Gpartial positive semidefinite matrix has a positive semidefinite matrix completion. In this paper, we relate these two conditions to constraint nondegeneracy condition in semidefinite programming and prove that they are each equivalent to (iii) For any Gpartial positive definite matrix that has a positive semidefinite completion, constraint nondegeneracy is satisfied at each of its positive semidefinite matrix completions.