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101.
Large scale optimization problems often require an approximation to the Hessian matrix. If the Hessian matrix is sparse then
estimation by differences of gradients is attractive because the number of required differences is usually small compared
to the dimension of the problem. The problem of estimating Hessian matrices by differences can be phrased as follows: Given
the sparsity structure of a symmetric matrixA, obtain vectorsd
1,d
2, …d
p such thatAd
1,Ad
2, …Ad
p determineA uniquely withp as small as possible. We approach this problem from a graph theoretic point of view and show that both direct and indirect
approaches to this problem have a natural graph coloring interpretation. The complexity of the problem is analyzed and efficient
practical heuristic procedures are developed. Numerical results illustrate the differences between the various approaches.
Work supported in part by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of
the U.S. Department of Energy under Contract W-31-109-Eng-38. 相似文献
102.
In this article we obtain a nonnegative rank factorization of nonnegative matrices A satisfying one or both of the following conditions: (i) AA ? ? 0 (ii) A ? A ? 0, thus providing a new set of conditions that guarantee the existence of a nonnegative least-squares solution of a linear system. Indeed, the characterization of such matrices improves some of the previous known conditions for the existence of a nonnegative least-squares solution of a linear system. 相似文献
103.
A. Mohammadian 《代数通讯》2013,41(12):4568-4574
We show that for any two n × n square-zero matrices A and B over a division ring, if the right column spaces of AB and BA are the same, then the rank of AB is at most n/4, and if, in addition, the right null spaces of AB and BA are the same, then the rank of A + B is at most n/2. This generalizes some known results. 相似文献
104.
We analyze the MAP/PH/1 vacation system at arbitrary times using the matrix-analytic method, and obtain decomposition results for the R and G matrices. The decomposition results reduce the amount of computational effort needed to obtain these matrices. The results for the G matrix are extended to the BMAP/PH/1 system. We also show that in the case of the Geo/PH/1 and M/PH/1 systems with PH vacations both the G and R matrices can be obtained explicitly. 相似文献
105.