Estimation of sparse hessian matrices and graph coloring problems
 Thomas F. Coleman,
 Jorge J. Moré
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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.
 T.F. Coleman and J.J. Moré, “Estimation of sparse Jacobian matrices and graph coloring problems”,SIAM Journal on Numerical Analysis 20 (1983), 187–209. CrossRef
 T.F. Coleman and J.J. Moré, “Software for estimating sparse Jacobian matrices”, Technical Report ANL8237, Argonne National Laboratory (Argonne, Illinois, 1982).
 A.R. Curtis, M.J.D. Powell and J.K. Reid, “On the estimation of sparse Jacobian matrices”,Journal of the Institute of Mathematics and its Applications 13 (1974) 117–119.
 S. Eisenstat, Personal communication (1980).
 G.C. Everstine, “A comparison of three resequencing algorithms for the reduction of matrix profile and wavefront”,International Journal on Numerical Methods in Engineering 14 (1979) 837–853. CrossRef
 M.R. Garey and D.S. Johnson, Computers and intractability (W.H. Freeman, San Francisco, CA, 1979).
 D.W. Matula, G. Marble and J.D. Isaacson, “Graph coloring algorithms”, in: R. Read, ed.,Graph theory and computing (Academic Press, New York, 1972), pp. 104–122.
 D.W. Matula and L.L. Beck, “Smallestlast ordering and clustering and graph coloring algorithms”,Journal of the Association for Computing Machinery 30 (1983) 417–427.
 S.T. McCormick, “Optimal approximation of sparse Hessians and its equivalence to a graph coloring problem”,Mathematical Programming 26 (1983), 153–171. CrossRef
 G.N. Newsam and J.D. Ramsdell, “Estimation of sparse Jacobian matrices”,SIAM Journal of Algebraic and Discrete Methods (1983), to appear.
 M.J.D. Powell and Ph.L. Toint, “On the estimation of sparse Hessian matrices”,SIAM Journal on Numerical Analysis 16 (1979) 1060–1074. CrossRef
 G. Szekeres and H.S. Wilf, “An inequality for the chromatic number of a graph”,Journal of Combinatorial Theory 4 (1968) 1–3.
 M.N. Thapa, “Optimization of unconstrained functions with sparse Hessian matrices: Newtontype methods”, Technical Report SOL 828, Systems Optimization Laboratory, Stanford University (Stanford, CA, 1982).
 Title
 Estimation of sparse hessian matrices and graph coloring problems
 Journal

Mathematical Programming
Volume 28, Issue 3 , pp 243270
 Cover Date
 19841001
 DOI
 10.1007/BF02612334
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Graph Coloring
 Estimation of Hessian Matrices
 Sparsity
 Differentiation
 Numerical Differences
 NPComplete Problems
 Unconstrained Minimization
 Industry Sectors
 Authors

 Thomas F. Coleman ^{(1)}
 Jorge J. Moré ^{(2)}
 Author Affiliations

 1. Department of Computer Science, Cornell University, 14853, Ithaca, NY, USA
 2. Mathematics and Computer Science Division, Argonne National Laboratory, 60439, Argonne, IL, USA