Global optimality conditions for discrete and nonconvex optimization-with applications to Lagrangian heuristics and column generation

Operations Research

Volumn 54, Issue 3, 2006, Pages 436-453

  Larsson, Torbjörn ^a   Patriksson, Michael ^b  

^a LINKÖPING UNIVERSITY   (Sweden)

^b CHALMERS UNIVERSITY OF TECHNOLOGY   (Sweden)

Author keywords

Column generation algorithms; Core problems; Programming, integer, algorithms: Lagrangian heuristics; Programming, integer, theory: global optimality conditions

Indexed keywords

ALGORITHMS; CONSTRAINT THEORY; HEURISTIC METHODS; OPTIMIZATION; PERTURBATION TECHNIQUES; PROBLEM SOLVING;

COLUMN GENERATION ALGORITHMS; CORE PROBLEMS; PROGRAMMING, INTEGER, ALGORITHMS: LAGRANGIAN HEURISTICS; PROGRAMMING, INTEGER, THEORY: GLOBAL OPTIMALITY CONDITIONS; SADDLE POINTS;

LAGRANGE MULTIPLIERS;

EID: 33744792363     PISSN: 0030364X     EISSN: 15265463     Source Type: Journal    
DOI: 10.1287/opre.1060.0292     Document Type: Article

References (58)

1. Ahuja, R. K., O. Ergun, J. B. Orlin, A. P. Punnen. 2002. A survey of very large scale neighborhood search techniques. Discrete Appl. Math. 123 75-102.
2. Andersen, E. D., J. E. Mitchell, C. Roos, T. Terlaky. 2001. A homogenized cutting plane method to solve the convex feasibility problem. X.-Q. Yang, K. L. Teo, L. Caccetta, eds. Optimization Methods and Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands, 167-190.
3. Appelgren, L. H. 1969. A column generation algorithm for a ship scheduling problem. Transportation Sci. 3 53-68.
4. Appelgren, L. H. 1971. Integer programming methods for a vessel scheduling problem. Transportation Sci. 5 64-78.
5. Balas, E., M. C. Carrera. 1996. A dynamic subgradient-based branch-andbound procedure for set covering. Oper. Res. 44 875-890.
6. Balas, E., A. Ho. 1980. An algorithm for set covering problems. Math. Programming Stud. 12 37-60.
7. Balas, E., E. Zemel. 1980. An algorithm for large zero-one knapsack problems. Oper. Res. 28 1130-1154.
8. Barnhart, C., E. L. Johnson, G. L. Nemhauser, M. W. P. Savelsbergh, P. H. Vance. 1998. Branch-and-price: Column generation for huge integer programs. Oper. Res. 46 316-329.
9. Bazaraa, M. S., H. D. Sherali, C. M. Shetty. 1993. Nonlinear Programming: Theory and Algorithms, 2nd ed. John Wiley and Sons, New York.
10. Beasley, J. E. 1987. An algorithm for set covering problems. Eur. J. Oper. Res. 31 85-93.
11. Beasley, J. E. 1993. Lagrangian relaxation. C. R. Reeves, ed. Chapter 6 in Modern Heuristic Techniques for Combinatorial Problems. Blackwell Scientific Publications, Oxford, UK, 243-303.
12. Bertsekas, D. P. 1999. Nonlinear Programming, 2nd ed. Athena Scientific, Belmont, MA.
13. Bertsekas, D. P., A. Nedić, A. E. Ozdaglar. 2003. Convex Analysis and Optimization. Athena Scientific, Belmont, MA.
14. Brown, G. G., A. Geoffrion, G. Bradley. 1981. Seasonal production and sales planning with limited shared tooling at the key operation. Management Sci. 27 247-259.
15. Campbell, G. M., V. A. Mabert. 1991. Cyclical schedules for capacitated lot sizing with dynamic demands. Management Sci. 17 409-427.
16. Caprara, A., M. Fischetti, P. Toth. 1999. A heuristic method for the set covering problem. Oper. Res. 47 730-743.
17. Caprara, A., M. Fischetti, P. Toth. 2002. Modeling and solving the train timetabling problem. Oper. Res. 50 851-861.
18. Carlyle, W. M., R. K. Wood. 2003. Lagrangian relaxation and enumeration for solving constrained shortest-path problems. Proc. 38th Annual ORSNZ Conf., University of Waikato, Hamilton, New Zealand, 3-12.
19. Ceria, S., P. Nobili, A. Sassano. 1998. A Lagrangian-based heuristic for large-scale set covering problems. Math. Programming 81 215-228.
20. Chvátal, V. 1979. A greedy heuristic for the set-covering problem. Math. Oper. Res. 4 233-235.
21. Erlenkotter, D. 1978. A dual-based procedure for uncapacitated facility location. Oper. Res. 26 992-1009.
22. Everett, III, H. 1963. Generalized Lagrange multiplier method for solving problem of optimal allocation of resources. Oper. Res. 11 399-518.
23. Feltenmark, S., K. C. Kiwiel. 2000. Dual applications of proximal bundle methods, including Lagrangian relaxation of nonconvex problems. SIAM J. Optim. 14 697-721.
24. Fisher, M. L. 1981. The Lagrangian relaxation method for solving integer programming problems. Management Sci. 27 1-18.
25. Fisher, M. L. 1985. An applications oriented guide to Lagrangian relaxation. Interfaces 15 10-21.
26. Fisher, M. L. 1994. Optimal solution of vehicle-routing problems using minimum K-trees. Oper. Res. 42 626-642.
27. Geoffrion, A. M. 1974. Lagrangian relaxation for integer programming. Math. Programming Stud. 2 82-114.
28. Handler, G. Y., I. Zang. 1980. A dual algorithm for the constrained shortest path problem. Networks 10 293-310.
29. Held, M., R. M. Karp. 1970. The traveling salesman problem and minimum spanning trees: Part I. Oper. Res. 18 1138-1162.
30. Held, M., R. M. Karp. 1971. The traveling-salesman problem and minimum spanning trees: Part II. Math. Programming 1 6-25.
31. Held, M., P. Wolfe, H. P. Crowder. 1974. Validation of subgradient optimization. Math. Programming 16 62-88.
32. Jonsson, Ö., T. Larsson, P. Varbrand. 1990. A Lagrangean relaxation scheme for a scheduling problem. Asia-Pacific J. Oper. Res. 7 155-162.
33. Kiwiel, K. C. 1995. Proximal level bundle methods for convex nondifferentiable optimization, saddle-point problems and variational inequalities. Math. Programming 69 89-109.
34. Larsson, T., Z.-W. Liu. 1997. A Lagrangean relaxation scheme for structured linear programs with application to multicommodity network flows. Optimization 40 247-284.
35. Larsson, T., M. Patriksson, A.-B. Strömberg. 1996. Conditional subgradient optimization - Theory and applications. Eur. J. Oper. Res. 88 382-403.
36. Larsson, T., M. Patriksson, A.-B. Strömberg. 1999. Ergodic, primal convergence in dual subgradient schemes for convex programming. Math. Programming 86 283-312.
37. Larsson, T., M. Patriksson, A.-B. Strömberg. 2003. On the convergence of conditional ε-subgradient methods for convex programs and convex-concave saddle-point problems. Eur. J. Oper. Res. 151 461-473.
38. Lasdon, L. S. 1970. Optimization Theory for Large Systems. Macmillan, New York.
39. Magnanti, T. L., J. F. Shapiro, M. H. Wagner. 1976. Generalized linear programming solves the dual. Management Sci. 21 1195-1203.
40. Marklund, J. 1993. A study of Lagrangian heuristics for convex network flow problems. Master's thesis, Department of Mathematics, Linköping Institute of Technology, Linköping, Sweden.
41. Martello, S., P. Toth. 1990. Knapsack Problems: Algorithms and Computer Implementations. Wiley, Chichester, UK.
42. Mulvey, J., H. Crowder. 1979. Cluster analysis: An application of Lagrangian relaxation. Management Sci. 25 329-340.
43. Nemhauser, G. L., L. A. Wolsey. 1988. Integer and Combinatorial Optimization, John Wiley and Sons, New York.
44. Patriksson, M. 1994. The Traffic Assignment Problem-Models and Methods. Topics in Transportation. VSP BV, Utrecht, The Netherlands.
45. Petersson, J., M. Patriksson. 1997. Topology optimization of sheets in contact by a subgradient method. Internat. J. Numer. Methods Engrg. 40 1295-1321.
46. Pisinger, D. 1995. An expanding-core algorithm for the exact 0-1 knapsack problem. Eur. J. Oper. Res. 87 175-187.
47. Polyak, B. T. 1969. Minimization of unsmooth functionals. USSR Comput. Math. Math. Phys. 9 14-29.
48. Savelsbergh, M. W. P. 1997. A branch-and-price algorithm for the generalized assignment problem. Oper. Res. 45 831-841.
49. Shapiro, J. F. 1979a. Mathematical Programming: Structures and Algorithms. John Wiley and Sons, New York.
50. Shapiro, J. F. 1979b. A survey of Lagrangian techniques for discrete optimization. Ann. Discrete Math. 5 113-138.
51. Sherali, H. D., G. Choi. 1996. Recovery of primal solutions when using subgradient optimization methods to solve Lagrangian duals of linear programs. Oper. Res. Lett. 19 105-113.
52. Strodiot, J.-J., V. H. Nguyen, N. Heukemes. 1983. ε-optimal solutions in nondifferentiable convex programming and some related questions. Math. Programming 25 307-328.
53. Sweeney, D. J., R. A. Murphy. 1979. A method of decomposition for integer programs. Oper. Res. 27 1128-1141.
54. Takkula, T. 2001. Private communication.
55. Vasko, F. J., G. R. Wilson. 1984. An efficient heuristic for large set covering problems. Naval Res. Logist. 31 163-171.
56. Ventura, J. A. 1991. Computational development of a Lagrangian dual approach for quadratic networks. Networks 21 469-485.
57. Wilhelm, W. E. 2001. A technical review of column generation in integer programming. Optim. Engrg. 2 159-200.
58. Wolsey, L. A. 1998. Integer Programming. John Wiley and Sons, New York.