On modified factorizations for large-scale linearly constrained optimization

SIAM Journal on Optimization

Volumn 9, Issue 4, 1999, Pages 1041-1063

  Gould, Nicholas Ian Mark ^a  

^a RUTHERFORD APPLETON LABORATORY   (United Kingdom)

Author keywords

Large scale problems; Linearly constrained optimization; Modified matrix factorizations; Second order optimality conditions

Indexed keywords

EID: 0033444426     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623495290660     Document Type: Article

References (41)

1. E. ANDERSON, Z. BAI, C. BISCHOF, J. DEMMEL, J. DONGARRA, J. DUCROZ, A. GREENBAUM, S. HAMMARLING, A. MCKENNEY, S. OSTROUCHOV, AND D. SORENSEN, LAPACK Users' Guide, 2nd ed., SIAM, Philadelphia, 1995.
2. M. ARIOLI, T. F. CHAN, I. S. DUFF, N. I. M. GOULD, AND J. K. REID, Computing a Search Direction for Large-Scale Linearly Constrained Nonlinear Optimization Calculations, Tech. Rep. RAL-93-066, Rutherford Appleton Laboratory, Chilton, Oxfordshire, England, 1993.
3. I. BONGARTZ, A. R. CONN, N. I. M. GOULD, AND PH. L. TOINT, CUTE: Constrained and unconstrained testing environment, ACM Trans. Math. Software, 21 (1995), pp. 123-160.
4. J. R. BUNCH AND L, C. KAUFMAN, Some stable methods for calculating inertia and solving symmetric linear equations, Math. Comp., 31 (1977), pp. 163-179.
5. J. R. BUNCH AND B. N. PARLETT, Direct methods for solving symmetric indefinite systems of linear equations, SIAM J. Numer. Anal., 8 (1971), pp. 639-655.
6. T. J. CARPENTER, I. J. LUSTIG, J. M. MULVEY, AND D. F. SHANNO, Higher-order predictor-corrector interior point methods with application to quadratic objectives, SIAM J. Optim., 3 (1993), pp. 696-725.
7. A. R. CONN, N. I. M. GOULD, AND PH. L. TOINT, LANCELOT: A Fortran Package for Large-Scale Nonlinear Optimization (Release A), Springer Ser. Comput. Math. 17, Springer-Verlag, Heidelberg, Berlin, New York, 1992.
8. R. W. COTTLE, Manifestations of the Schur complement, Linear Algebra Appl., 8 (1974), pp. 189-211.
9. J. E. DENNIS AND R. B. SCHNABEL, A view of unconstrained optimization, in Handbook of Operations Research and Management Science, vol. 1. Optimization, G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, eds., North-Holland, Amsterdam, 1989, pp. 1-72.
10. J. J. DONGARRA, I. S. DUFF, D. C. SORENSEN, AND H. A. VAN DER VORST, Solving Linear Systems on Vector and Shared Memory Computers, SIAM, Philadelphia, PA, 1991.
11. I. S. DUFF, A. M. ERISMAN, AND J. K. REID, Direct methods for sparse matrices, Oxford University Press, Oxford, UK, 1986.
12. I. S. DUFF, N. I. M. GOULD, J. K. REID, J, A. SCOTT, AND K. TURNER, The factorization of sparse symmetric indefinite matrices, IMA J. Numer. Anal., 11 (1991), pp. 181-204.
13. I. S. DUFF AND J. K. REID, MA27 : A Set of Fortran Subroutines for Solving Sparse Symmetric Sets of Linear Equations, Report R-10533, AERE Harwell Laboratory, Harwell, UK, 1982.
14. I. S. DUFF AND J. K. REID, The multifrontal solution of indefinite sparse symmetric linear equations, ACM Trans. Math. Software, 9 (1983), pp. 302-325.
15. I. S. DUFF AND J. K. REID, Exploiting zeros on the diagonal in the direct solution of indefinite sparse symmetric systems, ACM Trans. Math. Software, 22 (1996), pp. 227-257.
16. I. S. DUFF, J. K. REID, N. MUNKSGAARD, AND H. B. NEILSEN, Direct solution of sets of linear equations whose matrix is sparse, symmetric and indefinite, J. Inst. Math. Appl., 23 (1979), pp. 235-250.
17. A. V. FIACCO AND G. P. MCCORMlCK, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley, Chicester, New York, 1968; reprinted as Classics Appl. Math. 4, SIAM, Philadelphia, PA, 1990.
18. A. V. FIACCO AND G. P. MCCORMlCK, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley, Chicester, New York, 1968; reprinted as Classics Appl. Math. 4, SIAM, Philadelphia, PA, 1990.
19. R. FLETCHER, Factorizing symmetric indefinite matrices, Linear Algebra Appl., 14 (1976), pp. 257-272.
20. R. FLETCHER, Practical Methods of Optimization, 2nd ed., John Wiley, Chicester, New York, 1987.
21. A. FORSGREN, P. E. GILL, AND W. MURRAY, Computing modified Newton directions using a partial Cholesky factorization, SIAM J. Sci. Comput., 16 (1995), pp. 139-150.
22. A. L. FORSGREN AND W. MURRAY, Newton methods for large-scale linear equality-constrained minimization, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 560-587.
23. R. FOURER AND S. MEHROTRA, Solving symmetrical indefinite systems in an interior-point method for linear programming, Math. Programming, 62 (1993), pp. 15-39.
24. A. GEORGE AND J. W. H. LIU, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, Englewood Cliffs, NJ, 1981.
25. P. E. GILL, G. H. GOLUB, W. MURRAY, AND M. A. SAUNDERS, Methods for modifying matrix factorizations, Math. Comp., 28 (1974), pp. 505-535.
26. P. E. GILL AND W. MURRAY, Newton-type methods for unconstrained and linearly constrained optimization, Math. Programming, 7 (1974), pp. 311-350.
27. P. E. GILL, W. MURRAY, D. B. PONCELÉON, AND M. A. SAUNDERS, Preconditioners for indefinite systems arising in optimization, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 292-311.
28. P. E. GILL, W. MURRAY, AND M. H. WRIGHT, Practical Optimization, Academic Press, London, New York, 1981.
29. N. I. M. GOULD, On practical conditions for the existence and uniqueness of solutions to the general equality quadratic-programming problem, Math. Programming, 32 (1985), pp. 90-99.
30. N. I. M. GOULD, Constructing Appropriate Models for Large-Scale, Linearly-Constrained, Nonconvex, Nonlinear, Optimization Algorithms, Tech. Rep. RAL-TR 95-037, Rutherford Appleton Laboratory, Chilton, Oxfordshire, England, 1995.
31. HARWELL SUBROUTINE LIBRARY, A Catalogue of Subroutines (Release 10), Advanced Computing Department, Harwell Laboratory, Harwell, UK, 1990.
32. I. J. LUSTIG, R. E. MARSTEN, AND D. F. SHANNO, Computational experience with a primaldual interior point method for linear programming, Linear Algebra Appl., 152 (1991), pp. 191-222.
33. I. J. LUSTIG, R. E. MARSTEN, AND D. F. SHANNO, On implementing Mehrotra's predictor-corrector interior-point method for linear programming, SIAM J. Optim., 2 (1992), pp. 435-449.
34. H. M. MARKOWITZ, The elimination form of the inverse and its application to linear programming, Management Sci., 3 (1957), pp. 255-269.
35. S. MEHROTRA, On the implementation of a primal-dual interior point method, SIAM J. Optim., 2 (1992), pp. 575-601.
36. J. J. MORÉ AND D. C. SORENSEN, On the use of directions of negative curvature in a modified Newton method, Math. Programming, 16 (1979), pp. 1-20.
37. B. A. MURTAGH AND M. A. SAUNDERS, Large-scale linearly constrained optimization, Math. Programming, 14 (1978), pp. 41-72.
38. T. SCHLICK, Modified Cholesky factorizations for sparse preconditioners, SIAM J. Sci. Comput., 14 (1993), pp. 424-445.
39. R. B. SCHNABEL AND E. ESKOW, A new modified Cholesky factorization, SIAM J. Sci. Statist. Comput., 11 (1991), pp. 1136-1158.
40. G. W. STEWART, Modifying pivot elements in gaussian elimination, Math. Comp., 28 (1967), pp. 537-542.
41. R. J. VANDERBEI AND T. J. CARPENTER, Symmetrical indefinite systems for interior point methods, Math. Programming, 58 (1993), pp. 1-32.