On decomposition methods for a class of partially separable nonlinear programs

Mathematics of Operations Research

Volumn 33, Issue 1, 2008, Pages 119-139

  Demiguel, Victor ^a   Nogales, Francisco J ^b  

^a LONDON BUSINESS SCHOOL   (United Kingdom)

^b UNIVERSIDAD CARLOS III DE MADRID   (Spain)

Author keywords

Bilevel programming; Decomposition algorithms; Schur complement

Indexed keywords

ALGORITHMS;

BI-LEVEL DECOMPOSITIONS; BILEVEL PROGRAMMING; BILEVEL PROGRAMS; DECOMPOSITION ALGORITHMS; DECOMPOSITION METHODS; INTERIOR-POINT METHODS; LOCAL CONVERGENCES; NONCONVEX; NONLINEAR PROGRAMS; SCHUR COMPLEMENT; SCHUR COMPLEMENT MATRIXES; SUB-PROBLEMS;

DECOMPOSITION;

EID: 61349108214     PISSN: 0364765X     EISSN: 15265471     Source Type: Journal    
DOI: 10.1287/moor.1070.0282     Document Type: Article

References (36)

1. Alexandrov, N. M., M. Y. Hussaini, eds. 1997. Multidisciplinary Design Optimization: State of the Art. SIAM, Philadelphia.
2. Alexandrov, N. M., R. M. Lewis. 2000. Analytical and computational aspects of collaborative optimization. Technical Report TM-2000-210104, NASA, Hampton, VA.
3. Benders, J. F. 1962. Partitioning procedures for solving mixed variables programming problems. Numerische Mathematik 4 238-252.
4. Birge, J. R., F. Louveaux. 1997. Introduction to Stochastic Programming. Springer-Verlag, New York.
5. Braun, R. D. 1996. Collaborative optimization: An architecture for large-scale distributed design. Doctoral dissertation, Stanford University, Standford, CA.
6. Braun, R. D., I. M. Kroo. 1997. Development and application of the collaborative optimization architecture in a multidisciplinary design environment. N. M. Alexandrov, M. Y. Hussaini, eds. Multidisciplinary Design Optimization: State of the Art. SIAM, Philadelphia, 98-116.
7. Byrd, R. H., M. E. Hribar, J. Nocedal. 1999. An interior point algorithm for large-scale nonlinear programming. SIAM J. Optim. 9 877-900.
8. Chun, B. J., S. M. Robinson. 1995. Scenario analysis via bundle decomposition. Ann. Oper. Res. 56 39-63.
9. Cohen, G., B. Miara. 1990. Optimization with an auxiliary constraint and decomposition. SIAM J. Control Optim. 28(1) 137-157.
10. Colson, B., P. Marcotte, G. Savard. 2005. Bilevel programming: A survey. 4OR 3 87-107.
11. Cottle, R. W. 1974. Manifestations of the Schur complement. Linear Algebra Appl. 8 189-211.
12. Cramer, E. J., J. E. Dennis, P. D. Frank, R. M. Lewis, G. R. Shubin. 1994. Problem formulation for multidisciplinary optimization. SIAM J. Optim. 4(4) 754-776.
13. DeMiguel, V., W. Murray. 2000. An analysis of collaborative optimization methods. Eighth AIAA/USAF/NASA/ISSMO Sympos. Multi-disciplinary Anal, and Optim., Long Beach, CA, Paper 00-4720, AIAA, Reston, VA.
14. DeMiguel, V., W. Murray. 2006. A local convergence analysis of bilevel decomposition algorithms. Optim. Engrg. 7 99-133.
15. Dempe, S. 2002. Foundations of Bilevel Programming. Kluwer Academic Publishers, Boston.
16. El-Bakry, A. S., R. A. Tapia, T. Tsuchiya, Y. Zhang. 1996. On the formulation and theory of Newton interior-point method for nonlinear programming. J. Optim. Theory Appl. 89 507-541.
17. Fiacco, A. V., G. P. McCormick. 1968. Nonlinear Programming: Sequential Unconstrained Minimization Techniques. John Wiley and Sons, New York.
18. Forsgren, A., P. E. Gill. 1998. Primal-dual interior methods for nonconvex nonlinear programming. SIAM J. Optim. 8(4) 1132-1152.
19. Gay, D. M., M. L. Overton, M. H. Wright. 1997. A primal-dual interior method for nonconvex nonlinear programming. Technical Report 97-4-08, Computing Sciences Research, Bell Laboratories, Murray Hill, NJ.
20. Geoffrion, A. M. 1972. Generalized Benders decomposition. J. Optim. Theory Appl. 10(4) 237-260.
21. Gould, N. I. M., D. Orban, A. Sartenaer, P. L. Toint. 2001. Superlinear convergence of primal-dual interior point algorithms for nonlinear programming. SIAM J. Optim. 11(4) 974-1002.
22. Haftka, R. T., J. Sobieszczanski-Sobieski. 1997. Multidisciplinary aerospace design optimization: Survey of recent developments. Structural Optim. 14 1-23.
23. Helgason, T., S. W. Wallace. 1991. Approximate scenario solutions in the progressive hedging algorithm. Ann. Oper: Res. 31 425-444.
24. Martinez, H. H., Z. Parada, R. A. Tapia. 1995. On the characterization of q-superlinear convergence of quasi-Newton interior-point methods for nonlinear programming. Boletín de la Sociedad Matemática Mexicana 1 1-12.
25. Medhi, D. 1990. Parallel bundle-based decomposition for large-scale structured mathematical programming problems. Ann. Oper: Res. 22 101-127.
26. Raghunathan, A. U., L. T. Biegler. 2003. Interior point methods for mathematical programs with, complementarity constraints. Technical report, Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh.
27. Robinson, S. M. 1986. Bundle-based decomposition: Description and preliminary results. A. Prekopa, J. Szelezsan, B. Strazicky, eds. System Modelling and Optimization. Lecture Notes in Control and Information Sciences. Springer-Verlag, Berlin, 751-756.
28. Rockafellar, R. T., R. J.-B. Wets. 1991. Scenarios and policy aggregation in optimization under uncertainty. Math. Oper. Res. 16 119-147.
29. Ruszczynski, A. 1986. A regularized decomposition for minimizing a sum of polyhedral functions. Math. Programming 35 309-333.
30. Ruszczynski, A. 1995. On convergence of an augmented Lagrangian decomposition method for sparse convex optimization. Math. Oper. Res. 20(3) 634-656.
31. Shimizu, K., Y. Ishizuka, J. F. Bard. 1997. Nondifferrntiable and Two-Level Mathematical Programming. Kluwer Academic Publishers, Boston.
32. Tammer, K. 1987. The application of parametric optimization and imbedding for the foundation and realization of a generalized primal decomposition approach. J. Guddat, H. Jongen, B. Kummer, F. Nozicka, eds. Parametric Optimization and Related Topics, Mathematical Research, Vol. 35. Akademie-Verlag, Berlin, 376-386.
33. Van Slyke, R., R. J.-B. Wets. 1969. L-shaped linear programs with application to optimal control and stochastic programming. SIAM J. Appl. Math. 17 638-663.
34. Vanderbei, R. J., D. F. Shanno. 1997. An interior-point algorithm for nonconvex nonlinear programming. Technical Report SOR-97-21, Statistics and Operations Research, Princeton University, Princeton, NJ.
35. Vicente, L. N., S. J. Wright. 2002. Local convergence of a primal-dual method for degenerate nonlinear programming. Computational Optim. Appl. 22 311-328.
36. Yamashita, H., H. Yabe. 1996. Superlinear and quadratic convergence of some primal-dual interior point methods for constrained optimization. Math. Programming 75 377-397.