Matrix-free interior point method

Computational Optimization and Applications

Volumn 51, Issue 2, 2012, Pages 457-480

  Gondzio, Jacek ^a  

^a UNIVERSITY OF EDINBURGH   (United Kingdom)

Author keywords

Implicit preconditioner; Interior point methods; Iterative methods; Linear programming; Matrix free; Programming; Quadratic

Indexed keywords

COMPUTATIONAL RESULTS; COMPUTER MEMORIES; INTERIOR POINT METHODS; MATRIX VECTOR MULTIPLICATION; MATRIX-FREE; NEWTON EQUATION; NUMERICAL PROPERTIES; PRECONDITIONED ITERATIVE METHODS; PRECONDITIONERS; QUADRATIC; TWO-STEP APPROACH;

ITERATIVE METHODS; JACOBIAN MATRICES; LINEAR ALGEBRA; LINEAR PROGRAMMING; MATHEMATICAL PROGRAMMING;

LARGE SCALE SYSTEMS;

EID: 84862004900     PISSN: 09266003     EISSN: 15732894     Source Type: Journal    
DOI: 10.1007/s10589-010-9361-3     Document Type: Article

References (37)

1. Al-Jeiroudi, G., Gondzio, J.: Convergence analysis of inexact infeasible interior point method for linear optimization. J. Optim. Theory Appl. 141, 231-247 (2009)
2. Altman, A., Gondzio, J.: Regularized symmetric indefinite systems in interior point methods for linear and quadratic optimization. Optim. Methods Softw. 11-12, 275-302 (1999)
3. Andersen, E.D., Gondzio, J., Mészáros, C., Xu, X.: Implementation of interior point methods for large scale linear programming. In: Terlaky, T. (ed.) Interior Point Methods in Mathematical Programming, pp. 189-252. Kluwer Academic, Dordrecht (1996).
4. Bellavia, S.: An inexact interior point method. J. Optim. Theory Appl. 96, 109-121 (1998)
5. Benzi, M., Golub, G., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1-137 (2005) (Pubitemid 40771317)
6. Bergamaschi, L., Gondzio, J., Venturin, M., Zilli, G.: Inexact constraint preconditioners for linear system arising in interior point methods. Comput. Optim. Appl. 36, 137-147 (2007) (Pubitemid 46723396)
7. Bergamaschi, L., Gondzio, J., Zilli, G.: Preconditioning indefinite systems in interior point methods for optimization. Comput. Optim. Appl. 28, 149-171 (2004)
8. Castro, J., Cuesta, J.: Quadratic regularizations in an interior point method for primal block-angular problems. Math. Program. (2010). Published online 19 February 2010. doi:10.1007/s10107-010-0341-2
9. Chen, S., Donoho, D., Saunders, M.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20, 33-61 (1998) (Pubitemid 128689501)
10. Colombo, M., Gondzio, J.: Further development of multiple centrality correctors for interior point methods. Comput. Optim. Appl. 41, 277-305 (2008)
11. Curtis, F., Nocedal, J., Wächter, A.: A matrix-free algorithm for equality constrained optimization problems with rank-deficient Jacobians. SIAM J. Optim. 20, 1224-1249 (2009)
12. D'Apuzzo, M., De Simone, V., Di Serafino, D.: On mutual impact of numerical linear algebra and large-scale optimization with focus on interior point methods. Comput. Optim. Appl. 45, 283-310 (2010)
13. Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact Newton methods. SIAM J. Numer. Anal. 19, 400-408 (1982)
14. Dikin, I.I.: On the speed of an iterative process. Uprav. Sist. 12, 54-60 (1974)
15. Dollar, H., Gould, N., Schilders, W., Wathen, A.: Implicit-factorization preconditioning and iterative solvers for regularized saddle-point problems. SIAM J. Matrix Anal. Appl. 28, 170-189 (2006) (Pubitemid 46277539)
16. Duff, I.S., Erisman, A.M., Reid, J.K.: Direct Methods for Sparse Matrices. Oxford University Press, New York (1987).
17. Ferris, M., Munson, T.: Interior point methods for massive support vector machines. SIAM J. Optim. 13, 783-804 (2003) (Pubitemid 36971112)
18. Fletcher, R.: Conjugate gradient methods for indefinite systems. In: Watson, G. (ed.) Numerical Analysis Dundee 1975, pp. 73-89. Springer, Berlin, New York (1976).
19. Freund, R.W., Jarre, F.: A QMR-based interior-point algorithm for solving linear programs. Math. Program. 76, 183-210 (1997) (Pubitemid 127417010)
20. Friedlander, M.P., Tseng, P.: Exact regularization of convex programs. SIAM J. Optim. 18, 1326-1350 (2007)
21. Gill, P.E., Murray, W., Ponceleón, D.B., Saunders, M.A.: Preconditioners for indefinite systems arising in optimization. SIAM J. Matrix Anal. Appl. 13, 292-311 (1992)
22. Gill, P.E., Saunders, M.A., Shinnerl, J.R.: On the stability of Cholesky factorization for symmetric quasi-definite systems. SIAM J. Matrix Anal. Appl. 17, 35-46 (1996)
23. Gondzio, J., Toraldo, G. (eds.): Linear algebra issues arising in interior point methods. Comput. Optim. Appl. 36, 137-341 (2007). Special issue
24. Gould, N.I.M., Hribar, M.E., Nocedal, J.: On the solution of equality constrained quadratic problems arising in optimization. SIAM J. Sci. Comput. 23, 1375-1394 (2001)
25. Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. Frontiers in Applied Mathematics, vol. 16. SIAM, Philadelphia (1995).
26. Kojima, M., Megiddo, N., Mizuno, S.: A primal-dual infeasible-interior- point algorithm for linear programming. Math. Program. 61, 263-280 (1993)
27. Korzak, J.: Convergence analysis of inexact infeasible-interior-point- algorithm for solving linear programming problems. SIAM J. Optim. 11, 133-148 (2000) (Pubitemid 33235756)
28. Lu, Z., Monteiro, R.D.S., O'Neal, J.W.: An iterative solver-based infeasible primal-dual path-following algorithm for convex quadratic programming. SIAM J. Optim. 17, 287-310 (2006)
29. Lukšan, L., Vlček, J.: Indefinitely preconditioned inexact Newton method for large sparse equality constrained nonlinear programming problems. Numer. Linear Algebra Appl. 5, 219-247 (1998) (Pubitemid 128475839)
30. Oliveira, A.R.L., Sorensen, D.C.: A new class of preconditioners for large-scale linear systems from interior point methods for linear programming. Linear Algebra Appl. 394, 1-24 (2005)
31. Paige, C.C., Saunders, M.A.: LSQR: An algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Softw. 8, 43-71 (1982)
32. Rozlozník, M., Simoncini, V.: Krylov subspace methods for saddle point problems with indefinite preconditioning. SIAM J. Matrix Anal. Appl. 24, 368-391 (2002) (Pubitemid 36446414)
33. Rusten, T., Winther, R.: A preconditioned iterative method for saddle point problems. SIAM J. Matrix Anal. Appl. 13, 887-904 (1992)
34. Saunders, M.A.: Cholesky-based methods for sparse least squares: The benefits of regularization. In: Adams, L., Nazareth, J.L. (eds.) Linear and Nonlinear Conjugate Gradient-Related Methods. pp. 92-100. SIAM, Philadelphia (1996).
35. Silvester, D., Wathen, A.: Fast iterative solution of stabilised Stokes systems part ii: Using general block preconditioners. SIAM J. Numer. Anal. 31, 1352-1367 (1994)
36. Stewart, G.W.: On scaled projections and pseudoinverses. Linear Algebra Appl. 112, 189-193 (1989)
37. Wright, S.J.: Primal-Dual Interior-Point Methods. SIAM, Philadelphia (1997)