Skip to main content
SpringerLink
Log in

The omnipresence of Lagrange

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

Lagrangian relaxation is usually considered in the combinatorial optimization community as a mere technique, sometimes useful to compute bounds. It is actually a very general method, inevitable as soon as one bounds optimal values, relaxes constraints, convexifies sets, generates columns, etc. In this paper we review this method, from both points of view of theory (to dualize a given problem) and algorithms (to solve the dual by nonsmooth optimization).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Alizadeh, F. (1995). Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM Journal on Optimization, 5(1), 13–51.

    Article  Google Scholar 

  • Anstreicher, K., & Wolsey, L. A. (1993, in press). On dual solutions in subgradient optimization, Unpublished manuscript, CORE, Louvain-la-Neuve, Belgium. Mathematical Programming.

  • Babonneau, F., & Vial, J. P. (2007, in press). Vial, Accpm with a nonlinear constraint and an active set strategy to solve nonlinear multicommodity flow problems. Mathematical Programming.

  • Barahona, F., & Anbil, R. (2000). The volume algorithm: producing primal solutions with a subgradient method. Mathematical Programming, 87(3), 385–399.

    Article  Google Scholar 

  • Bertsekas, D. P. (1995). Nonlinear programming. Athena Scientific.

  • Bertsekas, D. P., Lauer, G. S., Sandell, N. R., & Posberg, T. A. (1983). Optimal short-term scheduling of large-scale power systems. IEEE Transactions on Automatic Control, 28, 1–11.

    Article  Google Scholar 

  • Briant, O., Lemaréchal, C., Meurdesoif, P., Michel, S., Perrot, N., & Vanderbeck, F. (2005, in press). Comparison of bundle and classical column generation. RR 5453, INRIA. http://www.inria.fr/rrrt/rr5453.html. Mathematical Programming.

  • Feltenmark, S., & Kiwiel, K. C. (2000). Dual applications of proximal bundle methods, including Lagrangian relaxation of nonconvex problems. SIAM Journal on Optimization, 10(3), 697–721.

    Article  Google Scholar 

  • Fisher, M. L. (1973). Optimal solution of scheduling problems using Lagrange multipliers: part I. Operations Research, 21, 1114–1127.

    Google Scholar 

  • Geoffrion, A. M. (1974). Lagrangian relaxation for integer programming. Mathematical Programming Study, 2, 82–114.

    Google Scholar 

  • Goemans, M. X. (1997). Semidefinite programming in combinatorial optimization. Mathematical Programming, 79, 143–161.

    Google Scholar 

  • Goemans, M. X., & Williamson, D. P. (1995). Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM, 6, 1115–1145.

    Article  Google Scholar 

  • Goffin, J.-L., Haurie, A., & Vial, J.-P. (1992). Decomposition and nondifferentiable optimization with the projective algorithm. Management Science, 38(2), 284–302.

    Article  Google Scholar 

  • Grinold, R. C. (1970). Lagrangian subgradients. Management Science, 17(3), 185–188.

    Google Scholar 

  • Grötschel, M., Lovász, L., & Schrijver, A. (1981). The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1, 169–197.

    Article  Google Scholar 

  • Held, M., & Karp, R. (1971). The traveling salesman problem and minimum spanning trees: part II. Mathematical Programming, 1(1), 6–25.

    Article  Google Scholar 

  • Hiriart-Urruty, J.-B., & Lemaréchal, C. (1993). Convex analysis and minimization algorithms. Heidelberg: Springer.

    Google Scholar 

  • Hiriart-Urruty, J.-B., & Lemaréchal, C. (2001). Fundamentals of convex analysis. Heidelberg: Springer.

    Google Scholar 

  • Horn, R. A., & Johnson, C. R. (1989). Matrix analysis. Cambridge: Cambridge University Press (new edition, 1999).

    Google Scholar 

  • Kiwiel, K. C. (1986). A method for solving certain quadratic programming problems arising in nonsmooth optimization. IMA Journal of Numerical Analysis, 6, 137–152.

    Article  Google Scholar 

  • Kiwiel, K. C. (2004, in press). An inexact bundle approach to cutting stock problems. Technical report, Systems Research Institute, Warsaw. INFORMS J. of Computing.

  • Kiwiel, K. C. (2006). A proximal bundle method with approximate subgradient linearizations. SIAM Journal on Optimization, 16(4), 1007–1023.

    Article  Google Scholar 

  • Kiwiel, K. C., & Lemaréchal, C. (2006, submitted). An inexact conic bundle variant suited to column generation. Open archive http://hal.inria.fr/inria-00109402. Mathematical Programming.

  • Larsson, T., Patriksson, M., & Strömberg, A. B. (1999). Ergodic, primal convergence in dual subgradient schemes for convex programming. Mathematical Programming, 86(2), 283–312.

    Article  Google Scholar 

  • Lasdon, L. (1970). Optimization theory for large systems. Macmillan series in operations research.

  • Lemaréchal, C. (1974). An algorithm for minimizing convex functions. In J. L. Rosenfeld (Ed.), Information processing ’74 (pp. 552–556). Amsterdam: North-Holland.

    Google Scholar 

  • Lemaréchal, C. (2001). Lagrangian relaxation. In M. Jünger, D. Naddef (Eds.), Computational combinatorial optimization (pp. 112–156). Heidelberg: Springer.

    Chapter  Google Scholar 

  • Lemaréchal, C. (2003). The omnipresence of Lagrange. 4OR, 1(1), 7–25.

    Article  Google Scholar 

  • Lemaréchal, C., Ouorou, A., & Petrou, G. (2006). A bundle-type algorithm for routing in telecommunication data networks. Technical report RR6010, INRIA. https://hal.inria.fr/inria-00110559.

  • Lemaréchal, C., & Oustry, F. (1999). Semi-definite relaxations and Lagrangian duality with application to combinatorial optimization. Rapport de Recherche 3710, INRIA. http://www.inria.fr/rrrt/rr-3710.html.

  • Lemaréchal, C., & Renaud, A. (2001). A geometric study of duality gaps, with applications. Mathematical Programming, 90(3), 399–427.

    Article  Google Scholar 

  • Lemaréchal, C., & Sagastizábal, C. (1997). Variable metric bundle methods: from conceptual to implementable forms. Mathematical Programming, 76(3), 393–410.

    Article  Google Scholar 

  • Magnanti, T. L., Shapiro, J. F., & Wagner, M. H. (1976). Generalized linear programming solves the dual. Management Science, 22(11), 1195–1203.

    Google Scholar 

  • Muckstadt, M. A., & Koenig, S. A. (1977). An application of Lagrangian relaxation to scheduling in power-generation systems. Operations Research, 25, 387–403.

    Google Scholar 

  • Nemirovskii, A. S., & Yudin, D. (1983). Problem complexity and method efficiency in optimization. Wiley-Interscience Series in Discrete Mathematics. New York: Wiley-Interscience (Original Russian: Moscow Nauka, 1979).

    Google Scholar 

  • Nurminskii, E. A., & Zhelikhovskii, A. A. (1977). ε-Quasigradient method for solving nonsmooth extremal problems. Cybernetics, 13(1), 109–114.

    Google Scholar 

  • Ouorou, A., Mahey, P., & Vial, J.-P. (2000). A survey of algorithms for convex multicommodity flow problems. Management Science, 47(1), 126–147.

    Article  Google Scholar 

  • Poljak, S., Rendl, F., & Wolkowicz, H. (1995). A recipe for semidefinite relaxation for (0,1)-quadratic programming. Journal of Global Optimization, 7, 51–73.

    Article  Google Scholar 

  • Reeves, C. R. (1993). Modern heuristic techniques for combinatorial problems. New York: Blackwell Scientific.

    Google Scholar 

  • Sagastizábal, C., Bahiense, L., & Maculan, N. (2002). The volume algorithm revisited: relation with bundle methods. Mathematical Programming, 94(1), 41–69.

    Article  Google Scholar 

  • Shor, N. Z. (1985). Minimization methods for non-differentiable functions. Berlin: Springer.

    Google Scholar 

  • Stetsenko, S. I., & Shor, N. Z. (1984). The connection between Lovász’ estimates with dual estimates in quadratic Boolean problems. In Solutions methods of nonlinear and discrete programming, proceedings of the seminar “National Council on Cybernetics”. Kiev: Institute of Cybernetics.

    Google Scholar 

  • Vandenberghe, L., & Boyd, S. (1996). Semidefinite programming. SIAM Review, 38(1), 49–95.

    Article  Google Scholar 

  • Vanderbeck, F. (2000). On Dantzig–Wolfe decomposition in integer programming and ways to perform branching in a branch-and-price algorithm. Operations Research, 48(1), 111–128.

    Article  Google Scholar 

  • Wolfe, P. (1975). A method of conjugate subgradients for minimizing nondifferentiable functions. Mathematical Programming Study, 3, 145–173.

    Google Scholar 

  • Wolsey, L. A. (1998). Integer programming. New York: Wiley-Interscience.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. 655 avenue de l’Europe, Montbonnot, 38s334, Saint Ismier, France

    Claude Lemaréchal

Authors
  1. Claude Lemaréchal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Claude Lemaréchal.

Additional information

This is an updated version of the paper that appeared in 4OR, 1(1), 7–25 (2003).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lemaréchal, C. The omnipresence of Lagrange. Ann Oper Res 153, 9–27 (2007). https://doi.org/10.1007/s10479-007-0169-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-007-0169-1

Keywords

Search

Navigation