Abstract
This paper is concerned with multilevel iterative methods which combine a descent scheme with a hierarchy of auxiliary problems in lower dimensional subspaces. The construction of auxiliary problems as well as applications to elasto-plastic model and linear programming are described. The auxiliary problem for the dual of a perturbed linear program is interpreted as a dual of perturbed aggregated linear program. Coercivity of the objective function over the feasible set is sufficient for the boundedness of the iterates. Equivalents of this condition are presented in special cases.
Similar content being viewed by others
References
M. Avriel,Nonlinear Programming: Analysis and Methods (Prentice-Hall, Englewood Cliffs, NJ, 1976).
E. Balas, “Solution of large-scale transportation problems through aggregation,”Operations Research 13 (1965) 82–93.
M.S. Bazaraa and C.M. Shetty,Nonlinear Programming: Theory and Algorithms (Wiley, New York, 1979).
R. Boyer and B. Martinet, “Multigrid methods in convex optimization,” in: U. Trotenberg and W. Hackbusch, eds.,Multigrid Methods: Special Topics and Applications. GMD Studien, Vol. 110 (Gesellschaft für Mathematik und Datenverarbeitung, Bonn, 1986) pp. 27–37.
A. Brandt, “Multi-level adaptive solutions to boundary value problems,”Mathematics of Computation 31 (1977) 333–390.
A. Brandt, “Multi-level computations: Review and recent developments,” in: S. McCormick, with J. Dendy, J. Mandel, S. Parter and J. Ruge, eds.,Multigrid Methods, Proceedings of the Third Copper Mountain Conference (Dekker, New York, 1988) pp. 35–62.
A. Brandt and C.W. Cryer, “Multigrid algorithms for the solution of linear complementarity problems arising from free boundary problems,”SIAM Journal on Scientific and Statistical Computing 4 (1983) 655–684.
W. Briggs,Multigrid Tutorial (SIAM, Philadelphia, PA, 1987).
F. Chatelin and W.L. Miranker, “Aggregation of successive approximation methods,”Linear Algebra and Applications 43 (1982) 17–47.
R.W. Cottle and J.S. Pang, “On the convergence of a block successive over-relaxation method for a class of linear complementarity problems,”Mathematical Programming Study 17 (1982) 126–138.
R. Glowinski, J.L. Lions and R. Trémolières,Numerical Analysis of Variational Inequalities (North-Holland, Amsterdam, 1981).
W. Hackbusch, “Convergence of multi-grid iterations applied to difference equations,”Mathematics of Computation 34 (1980) 425–440.
W. Hackbusch,Multigrid Methods and Applications (Springer, Berlin, 1985).
W. Hackbusch and H.D. Mittelmann, “On multigrid methods for variational inequalities,”Numerische Mathematik 42 (1983) 65–76.
W. Hackbusch and U. Trottenberg, “Multigrid methods,”Proceedings, Lecture Notes in Mathematics, Vol. 960 (Springer, Berlin, 1982).
W. Hackbusch and W. Trottenberg, “Multigrid methods II,”Proceedings, Lecture Notes in Mathematics, Vol. 1228 (Springer, Berlin, 1985).
G. Liesegang, “Aggregation bei linearen optimierungs-modellen,” Habilitationsschrift, Universität zu Köln (Cologne, 1980.
J. Mandel, “Multilevel iterative methods for some variational inequalities and optimization problems,” Technical Report 32, Computing Center, Charles University (Prague, Czechoslovakia, 1983).
J. Mandel, “Algebraic study of a multigrid method for some free boundary problems,”Comptes Rendus Academic of Science Paris, Series I 298 (1984) 469–472.
J. Mandel, “A multilevel iterative method for symmetric, positive definite linear complementarity problems,”Applied Mathematics and Optimization 11 (1984) 77–95.
J. Mandel and B. Sekerka, “A local convergence proof for the iterative aggregation method,”Linear Algebra and Applications 51 (1983) 163–172.
O.L. Mangasarian, “Solution of symmetric linear complementarity problems by iterative methods,”Journal of Optimization Theory and Applications 22 (1977) 465–485.
O.L. Mangasarian, “Iterative solution of linear programs,”SIAM Journal on Numerical Analysis 18 (1981) 606–614.
O.L. Mangasarian and R.R. Meyer, “Nonlinear perturbations of linear programs,”SIAM Journal for Control and Optimization 17 (1979) 745–752.
S.F. McCormick,Multigrid Methods (SIAM, Philadelphia, PA, 1987).
W.L. Miranker and V. Ya. Pan, “Methods of aggregation,”Linear Algebra and Applications 29 (1980) 231–257.
R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
I.Y. Vakhutinsky, L.M. Dudkin and A.A. Ryvkin, “Iterative aggregation—A new approach to the solution of large-scale problems,”Econometrica 47 (1979) 821–841.
W.I. Zangwill,Nonlinear Programming—A Unified Approach (Prentice-Hall, Englewood Cliffs, NJ, 1969).
P. Zipkin, “Bounds for row-aggregation in linear programming,”Operations Research 28 (1980) 903–916.
P. Zipkin, “Bounds on the effect of aggregating variables in linear programs,”Operations Research 28 (1980) 403–418.
Author information
Authors and Affiliations
Additional information
Supported by NSF under grant DMS-8704169, AFOSR under grant 86-0126, and ONR under Contract N00014-83-K-0104. Consulting for American Airlines Decision Technologies, MD 2C55, P.O. Box 619616, DFW, TX 75261-9616, USA.
Supported by NSF under grant DMS-8704169 and AFOSR under grant 86-0126.
Rights and permissions
About this article
Cite this article
Gelman, E., Mandel, J. On multilevel iterative methods for optimization problems. Mathematical Programming 48, 1–17 (1990). https://doi.org/10.1007/BF01582249
Issue Date:
DOI: https://doi.org/10.1007/BF01582249