
	Where results make sense

Topic: Positive semidefinite

In the News (Wed 30 Dec 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	PlanetMath: positive definite
		Thus the determinant of a positive definite matrix is positive, and a positive definite matrix is always invertible.
		Further conditions and properties for positive definite matrices are given in [2].
		This is version 6 of positive definite, born on 2002-02-15, modified 2004-06-17.
	planetmath.org /encyclopedia/PositiveDefinite.html (237 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		a pivoting option allows the user to estimate the c condition of a positive definite matrix or determine the rank c of a positive semidefinite matrix.
		c c for positive definite matrices info = p is the normal return.
		c for pivoting with positive semidefinite matrices info will c in general be less than p.
	www.cs.uiowa.edu /~luke/classes/248/R-1.8.1/src/appl/dchdc.f (844 words)

	Positive-definite matrix - Wikipedia, the free encyclopedia
		In linear algebra, a positive-definite matrix is a Hermitian matrix which in many ways is analogous to a positive real number.
		The notion is closely related to a positive-definite symmetric bilinear form (or a sesquilinear form in the complex case).
		Every positive definite matrix is invertible and its inverse is also positive definite.
	en.wikipedia.org /wiki/Positive-definite_matrix (447 words)

	Solutions to exercises on quadratic forms (Site not responding. Last check: 2007-10-21)
		Thus the form is indefinite: one of the first-order principal minors is positive, but the second-order one that is obtained by deleting the third row and column of the matrix is negative.
		Thus for a = 0 the matrix is positive semidefinite, and for other values of a the matrix is indefinite.
		The matrix is not positive definite or positive semidefinite for any values of a and b, because two of the first-order principal minors are negative.
	www.chass.utoronto.ca /~osborne/MathTutorial/QFSX1S.HTM (414 words)

	Fields Institute - Combinatorial Optimization Problems
		Semidefinite programming is well known to provide powerful relaxations for quadratic 0/1 programming and, as we intend to show, it is very useful for quadratic knapsack problems as well.
		We extend this notion to semidefinite programming and use it to prove characterizations of the distance to ill-posedness of a given data instance, upper and lower bounds on sizes of solutions, and rates of change of solutions along the central trajectory of a semidefinite program.
		Semidefinite programming relaxations for the quadratic assignment problem QAP are derived using the dual of the Lagrangian dual of appropriate equivalent representations of QAP.
	www.fields.utoronto.ca /programs/scientific/95-96/optimization/abstracts.html (4723 words)

	DIMACS Conference of SDP: Program
		Abstract: We consider the general semidefinite optimization problem: Compute the infimum of a linear objective function of an $n \times n$ symmetric positive semidefinite matrix satisfying a given system of $m$ linear constraints; if the infimum is attained, find the least-norm optimal solution.
		For randomly generated semidefinite programs, not only do the nondegeneracy conditions hold with probability one, but experiments show that the ranks of the solution matrices fall in the middle of the range permitted by the nondegeneracy conditions with surprisingly high probability.
		Semidefinite programming has been used to find significantly improved approximation algorithms (in terms of nearness to optimality) for the maximum cut, maximum satisfiability, quadratic programming, and graph coloring problems, as well as some others.
	www.sor.princeton.edu /~rvdb/sdp_sched.html (4362 words)

	Quadratic forms: conditions for semidefiniteness
		As in the case of two variables, to determine whether a quadratic form is positive or negative semidefinite we need to check more conditions than we do in order to check whether it is positive or negative definite.
		In particular, it is not true that a quadratic form is positive or negative semidefinite if the inequalities in the conditions for positive or negative definiteness are satisfied weakly.
		Neither the conditions for A to be positive definite nor those for A to be negative definite are satisfied.
	free.prohosting.com /cepr/data/adveco/qfs.html (895 words)

	Publications (in mathematics) list for Henry Wolkowicz
		Semidefinite programming (SDP) relaxations for the quadratic assignment problem (QAP) are derived using the dual of the (homogenized) Lagrangian dual of appropriate equivalent representations of QAP.
		Semidefinite linear programming (SDP) is a generalization of LP where the nonnegativity constraints are replaced by a semidefiniteness constraint on the matrix variables.
		The semidefinite framework is studied as an interesting instance of semidefinite programming as well as a tool for viewing known algorithms and deriving new algorithms for TRS.
	orion.math.uwaterloo.ca /~hwolkowi/henry/reports/ABSTRACTS.html (11292 words)

	[No title]
		Thus A should be positive semidefinite (it will have 0 as an eigenvalue if the whole system, or part of it, is free to move in some direction).
		This is a generic small-scale behavior of a smooth function (of many variables) near a local minimum, and positive definiteness of the matrix of the second derivatives (this matrix is called the Hessian) is a part of the test for a local minimum.
		In mechanics, the positive definiteness of the Hessian near a stationary point of the potential indicates a stable equilibrium.
	www.math.niu.edu /~rusin/known-math/99/posdef (784 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		The Covariance Matrix is Positive Semidefinite Any covariance matrix is positive semidefinite:-- Let S be a covariance matrix of a random vector X. Let m be the mean vector of X. Let E denote Expected Value.
		This means v'(X-m) = 0, or v'X = v'm; i.e., there exists a linear combination of the elements of X which is equal to its mean with probability 1, that is to say, there is a variate which is degenerate in this sense of being a constant random variable.
		Except when this happens, the covariance matrix is positive definite, not just positive semidefinite.
	www.uic.edu /classes/bstt/bstt580/pdcovmx.txt (137 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		To obtain the feasible ML estimates, an optimization problem whose objective is a nonlinear function of a positive semidefinite matrix variable must be solved.
		Semidefinite programming has attracted much research in the last few years, but most of the work has addressed problems where the problem is “linear.” Because the ML formulation of the problem results in a nonlinear function, a nonlinear semi-definite program must be solved.
		However, the nonlinear semidefinite programming method developed (see Reference [8]) and discussed here, also finds a ML estimate of the covariance matrix that is positive semidefinite; and, as the computational results show, the nonlinear semidefinite programming approach is faster than Bresler’s approach.
	www.mitretek.org /pubs/telecom/review01/11Paper9_Smith.doc (1767 words)

	Henry Wolkowicz: Optimization Software and Theory, e.g. Semidefinite Programming (Site not responding. Last check: 2007-10-21)
		Semidefinite programs are linear programs where the nonnegativity constraint is replaced by a positive semidefinite constraint on matrix variables.
		Semidefinite programs arise in many applications, e.g., combinatorial optimization, control theory, statistics, and nonlinear programming.
		Semidefinite Programming Applied to Nonlinear Programming Master's Thesis by Serge Kruk, a preliminary to an SQQP algorithm, and a Research Report on an SQQP algorithm.
	orion.math.uwaterloo.ca /~hwolkowi/henry/software/readme.html (1924 words)

	ISMP 2000 - Meeting Topics (Site not responding. Last check: 2007-10-21)
		Based on some fundamental results about positive semidefinite matrix completion, and the so related chordal graphs, we will propose a general method of exploiting the aggregate sparsity pattern over all data matrices to overcome this disadvantage.
		One is a conversion of a sparse SDP having a large scale positive semidefinite variable matrix into an SDP having multiple but smaller size positive semidefinite variable matrices to which we can apply the currently available software for SDPs.
		Based on the general framework for exploiting sparsity in semidefinite programming via matrix completion, we discuss practical implementation of a primal-dual interior-point method using matrix completion, and report numerical results on some types of large scale problems to evaluate the effectiveness and computational efficiency of the method.
	www.isye.gatech.edu /ismp2000/schedule/session_pages/TUC-17-IC209.html (328 words)

	A Decomposition Method for Positive Semidefinite Matrices and Its Application to Recursive Parameter Estimation
		A Decomposition Method for Positive Semidefinite Matrices and Its Application to Recursive Parameter Estimation: SIAM Journal on Matrix Analysis and Applications Vol.
		A matrix decomposition method for positive semidefinite matrices based on a given subspace is proposed in this paper.
		It is shown that any positive semidefinite matrix can be decomposed uniquely into two positive semidefinite parts with specified rank one of which is orthogonal to the subspace.
	epubs.siam.org /sam-bin/dbq/article/36402 (184 words)

	ipedia.com: Density state Article (Site not responding. Last check: 2007-10-21)
		It is the quantum-mechanical analogue to a phase-space density (probability distribution of position and momentum) in classical statistical mechanics.
		The need for a statistical description via density matrices arises because it is not possible to describe a quantum mechanical system that undergoes general quantum operations such as measurement, using exclusively states represented by ket vectors.
		For the density matrix, this means that ρ is a positive semidefinite hermitian operator (its eigenvalues are nonnegative) and the trace of ρ (the sum of its eigenvalues) is equal to one.
	www.ipedia.com /density_state.html (407 words)

	Quadratic forms: conditions for definiteness
		, but if the conditions for positive definiteness are satisfied then it must in fact also be true that c > 0 (and similarly c < 0 in the case of negative definiteness).
		To obtain conditions for an n-variable quadratic form to be positive or negative definite, we need to examine the determinants of some of its submatrices.
		The following result characterizes positive and negative definite quadratic forms (and their associated matrices).
	free.prohosting.com /cepr/data/adveco/qff.html (727 words)

	Top: SemidefiniteProgramming
		Semidefinite programming is a generalization of linear programming.
		A semidefinite program (SDP) is a linear program, along with constraints that say that some of the variables should form a positive semidefinite matrix.
		A corollary to this theorem is that semidefinite programs can be solved using the EllipsoidMethod in polynomial time.
	www.cs.ucr.edu /~neal/wiki/wiki.pl?action=browse&id=SemidefiniteProgramming&revision=3 (512 words)

	Exercises on quadratic forms
		Determine whether each of the following quadratic forms in two variables is positive or negative definite or semidefinite, or indefinite.
		is positive definite, negative definite, positive semidefinite, negative semidefinite, and indefinite.
		Find conditions on a and b under which this matrix is negative definite, negative semidefinite, positive definite, positive semidefinite, and indefinite.
	www.chass.utoronto.ca /~osborne/MathTutorial/QFSX1.HTM (114 words)

	Semidefinite Programming (Site not responding. Last check: 2007-10-21)
		ABSTRACT: In semidefinite programming we minimize a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite.
		Semidefinite programming unifies several standard problems (\eg, linear and quadratic programming) and finds many applications in engineering.
		Although semidefinite programs are much more general than linear programs, they are just as easy to solve.
	www.stanford.edu /~boyd/sdp.html (144 words)

	Quadratic forms: conditions for definiteness
		Thus, to determine whether a quadratic form is positive or negative definite we need to look only at the signs of a and of ac
		, but if the conditions for positive definiteness are satisfied then it must in fact also be true that c > 0, and if the conditions for negative definitely are satisfied then we must also have c < 0.
		Thus A is neither positive definite nor negative definite.
	www.chass.utoronto.ca /~osborne/MathTutorial/QFF.HTM (654 words)

	Cuts, Matrix Completions and Graph Rigidity (Site not responding. Last check: 2007-10-21)
		We consider several topics arising in distinct areas: cut and metric polyhedra in polyhedral combinatorics, completion problems for positive semidefinite matrices and Euclidean distance matrices in matrix theory and semidefinite programming, and graph realization and rigidity problems in distance geometry and structural topology.
		Indeed, cuts can be encoded as positive semidefinite matrices and both positive semidefinite and Euclidean distance matrices yield points of the cut polytope or cone, after applying the functions ${1\over \pi} \arccos (.)$ or $\sqrt{} $.
		The matrix completion problem asks whether the unspecified entries of a partially defined matrix can be completed so as to yield a matrix satisfying a desired property, like being a positive semidefinite matrix or a Euclidean distance matrix.
	dmawww.epfl.ch /roso.mosaic/ismp97/ismp_abs_506.html (252 words)

	Semidefinite Programming in the Space of Partial Positive Semidefinite Matrices
		Semidefinite Programming in the Space of Partial Positive Semidefinite Matrices: SIAM Journal on Optimization Vol.
		303--327], in which the theory of partial positive semidefinite matrices was applied to the semidefinite programming (SDP) problem as a technique for exploiting sparsity in the data.
		In contrast to their work, which improved an existing algorithm based on a standard search direction, we present a primal-dual path-following algorithm that is based on a new search direction, which, roughly speaking, is defined completely within the space of partial symmetric matrices.
	epubs.siam.org /sam-bin/dbq/article/40851 (207 words)

	On Positive Semidefinite Matrices with Known Null Space
		On Positive Semidefinite Matrices with Known Null Space: SIAM Journal on Matrix Analysis and Applications Vol.
		We show how the zero structure of a basis of the null space of a positive semidefinite matrix can be exploited to determine a positive definite submatrix of maximal rank.
		We furthermore execute a backward error analysis of the Cholesky factorization of positive semidefinite matrices and provide new elementwise bounds.
	epubs.siam.org /sam-bin/dbq/article/38133 (155 words)

	Monique Laurent
		Lecture on Semidefinite Programming in Polynomial Optimization at the SIAM Conference on Optimization, Stockholm, May 15-19, 2005.
		Semidefinite bounds for the stability number of a graph via sums of squares of polynomials.
		Lower bound for the number of iterations in semidefinite relaxations for the cut polytope.
	homepages.cwi.nl /~monique (817 words)

	Semidefinite Programming
		The (linear) semidefinite programming problem (SDP) is essentially an ordinary linear program where the nonnegativity constraint is replaced by a semidefinite constraint on matrix variables.
		Both the semidefinite cone (for SDP) and the non-negative orthant (for LP) are homogeneous, self-dual cones - there are only 5 such nonisomorphic categories of cones.
		One of the main aspects in which SDP differs from LP is that the non-negative orthant is a polyhedral cone, whereas the semidefinite cone is not.
	www-fp.mcs.anl.gov /otc/Guide/OptWeb/continuous/constrained/sdp (1042 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Abstract: An n-tuple A_1...A_n of n by n positive semidefinite matrices is 'doubly-stochastic' if they sum to identity, and the trace of each matrix is one.
		It turns out that, under some mild conditions, each n-tuple of positive semidefinite matrices could be 'scaled' to a doubly stochastic tuple.
		Using this, we prove the following theorem: Let A_1...A_n be positive semidefinite matrices, and r_1 \ge r_2 \ge...
	www.math.technion.ac.il /~techm/20011128130020011128sam (263 words)

	Positive Semidefinite Matrices (Site not responding. Last check: 2007-10-21)
		i.e., the matrix A is a positive semidefinite matrix, but it is not positive definite.
		Since the matrix A is positive semidefinite and symmetric, then
		is applicable (with the small changes) also to the symmetric positive semidefinite matrix A.
	www.cs.ut.ee /~toomas_l/linalg/lin2/node25.html (88 words)

	00-34 (Site not responding. Last check: 2007-10-21)
		We equip the reader with the basic results from linear algebra on positive semidefinite matrices and the cone spanned by them.
		We then turn to semidefinite relaxations of combinatorial optimization and illustrate their interrelation.
		One of the most successful techniques in integer linear programming is the cutting plane approach which improves an initial relaxation by adding violated inequalities.
	www.zib.de /PaperWeb/abstracts/ZR-00-34 (194 words)

	Brian's Digest: Semidefinite Programming (Site not responding. Last check: 2007-10-21)
		A.X = b X is positive semidefinite (D) max b y s.t.
		C-yA is positive semidefinite Q: Note that (P) has n^2 variables and (D) has only one variable.
		Is there a ratio (dependent on the dimension) of the volume of the positive semidefinite matrices relative to the symmetric matrices?
	www.worms.ms.unimelb.edu.au /digest/semidefinite_pr.html (316 words)

	Steven Benson Seminar of 19 Feb 1998 (Site not responding. Last check: 2007-10-21)
		The positive semidefinite relaxation can be an effective technique for approximating solutions to combinatorial optimization problems.
		It is essential, however, to exploit the structure and sparsity characteristic of most large-scale problems to quickly solve the semidefinite program.
		Coupled with a randomized technique that uses the solution matrix to identify high quality approximations, this algorithm has been successfully tested on problems with dimension of over 3000.
	www-fp.mcs.anl.gov /division/information/seminars/1998/benson_980219.htm (83 words)

Try your search on: Qwika (all wikis)