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Computational Combinatorial Optimization pp 112–156Cite as

Lagrangian Relaxation

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2241))

Abstract

Lagrangian relaxation is a tool to find upper bounds on a given (arbitrary) maximization problem. Sometimes, the bound is exact and an optimal solution is found. Our aim in this paper is to review this technique, the theory behind it, its numerical aspects, its relation with other techniques such as column generation.

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References

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

    MathSciNet  MATH  Google Scholar 

  2. K. Anstreicher and L.A. Wolsey. On dual solutions in subgradient opimization. Unpublished manuscript, CORE, Louvain-la-Neuve, Belgium, 1993.

    Google Scholar 

  3. D.P. Bertsekas. Projected Newton methods for optimization problems with simple constraints. SIAM Journal on Control and Optimization, 20:221–246, 1982.

    MathSciNet  MATH  Google Scholar 

  4. M.J. Best. Equivalence of some quadratic programming algorithms. Mathematical Programming, 30:71–87, 1984.

    MathSciNet  MATH  Google Scholar 

  5. U. Brännlund. On relaxation methods for nonsmooth convex optimization. PhD thesis, Royal Institute of Technology-Stockholm, 1993.

    Google Scholar 

  6. P.M. Camerini, L. Fratta, and F. Maffioli. On improving relaxation methods by modified gradient techniques. Mathematical Programming Study, 3:26–34, 1975.

    MathSciNet  MATH  Google Scholar 

  7. E. Cheney and A. Goldstein. Newton’s method for convex programming and Tchebyche. approximations. Numerische Mathematik, 1:253–268, 1959.

    MathSciNet  MATH  Google Scholar 

  8. R. Correa and C. Lemaréchal. Convergence of some algorithms for convex minimization. Mathematical Programming, 62(2):261–275, 1993.

    MathSciNet  MATH  Google Scholar 

  9. A. Decarreau, D. Hilhorst, C. Lemarechal, and J. Navaza. Dual methods in entropy maximization. application to some problems in crystallography. SIAM Journal on Optimization, 2(2):173–197, 1992.

    MathSciNet  MATH  Google Scholar 

  10. I. Ekeland and R. Temam. Convex Analysis and Variational Problems. North Holland, 1976; reprinted by SIAM, 1999.

    Google Scholar 

  11. Y.M. Ermol'ev. Methods of solution of nonlinear extremal problems. Kibernetica, 2(4):1–17, 1966.

    Google Scholar 

  12. H. Everett III. Generalized lagrange multiplier method for solving problems of optimum allocation of resources. Operations Research, 11:399–417, 1963.

    MathSciNet  MATH  Google Scholar 

  13. J.E. Falk. Lagrange multipliers and nonconvex programs. SIAM Journal on Control, 7(4):534–545, 1969.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  15. R. Fletcher. Practical Methods of Optimization. John Wiley & Sons, Chichester (second edition), 1987.

    MATH  Google Scholar 

  16. A. Frangioni. Solving semidefinite quadratic problems within nonsmooth optimization algorithms. Computational Operational Research, 23(11):1099–1118, 1996.

    MathSciNet  MATH  Google Scholar 

  17. M. Fukushima and L. Qi. A globally and superlinearly convergent algorithm for nonsmooth convex minimization. SIAM Journal on Optimization, 6:1106–1120, 1996.

    MathSciNet  MATH  Google Scholar 

  18. C. Garrod and J.K. Percus. Reduction of the N-particle variational problem. Journal of Mathematical Physics, 5(12), 1964.

    Google Scholar 

  19. A.M. Geoffrion. Lagrangean relaxation for integer programming. Mathematical Programming Study, 2:82–114, 1974.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  21. J.L. Goffin, A. Haurie, and J.Ph. Vial. Decomposition and nondifferentiable optimization with the projective algorithm. Management Science, 38(2):284–302, 1992.

    MATH  Google Scholar 

  22. J.L. Goffin and J.Ph. Vial. Convex nondifferentiable optimization: a survey focussed on the analytic center cutting plane method. to appear in Optimization Methods and Software; also as HEC/Logilab Technical Report 99.02, Univ. of Geneva, Switzerland.

    Google Scholar 

  23. M. Held and R. Karp. The travelling salesman problem and minimum spanning trees: Part II. Mathematical Programming, 1:6–25, 1971.

    MathSciNet  MATH  Google Scholar 

  24. C. Helmberg and F. Rendl. A spectral bundle method for semidefinite programming. SIAM Journal on Optimization, 10(3):673–696, 2000.

    MathSciNet  MATH  Google Scholar 

  25. J.-B. Hiriart-Urruty and C. Lemaréchal. Convex Analysis and Minimization Algorithms. Springer Verlag, Heidelberg, 1993.

    MATH  Google Scholar 

  26. J.-B. Hiriart-Urruty and C. Lemaréchal. Fundamentals of Convex Analysis. Springer Verlag, Heidelberg, 2001. to appear.

    MATH  Google Scholar 

  27. R.A. Horn and Ch.R. Johnson. Matrix Analysis. Cambridge University Press, 1989. (New edition, 1999).

    Google Scholar 

  28. J. E. Kelley. The cutting plane method for solving convex programs. J. Soc. Indust. Appl. Math., 8:703–712, 1960.

    MathSciNet  MATH  Google Scholar 

  29. K.C. Kiwiel. Methods of Descent for Nondifferentiable Optimization. Lecture Notes in Mathematics 1133. Springer Verlag, Heidelberg, 1985.

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  31. K.C. Kiwiel. A dual method for certain positive semidefinite quadratic programming problems. SIAM Journal on Scientific and Statistical Computing, 10:175–186, 1989.

    MathSciNet  MATH  Google Scholar 

  32. K.C. Kiwiel. Proximity control in bundle methods for convex nondifferentiable minimization. Mathematical Programming, 46(1):105–122, 1990.

    MathSciNet  MATH  Google Scholar 

  33. K.C. Kiwiel. Efficiency of proximal bundle methods. Journal of Optimization Theory and Applications, 104(3):589–603, 2000.

    MathSciNet  MATH  Google Scholar 

  34. K.C. Kiwiel, T. Larsson, and P.O. Lindberg. The efficiency of ballstep subgradient level methods for convex optimization. Mathematics of Operations Research, 24(1):237–254, 1999.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  36. C. Lemaréchal, A.S. Nemirovskii, and Yu.E. Nesterov. New variants of bundle methods. Mathematical Programming, 69:111–148, 1995.

    MathSciNet  MATH  Google Scholar 

  37. C. Lemaréchal, Yu. Nesterov, and F. Oustry. Duality gap analysis for problems with quadratic constraints, 2001. In preparation.

    Google Scholar 

  38. C. Lemaréchal and F. Oustry. Semi-definite relaxations and lagrangian duality with application to combinatorial optimization. Rapport de Recherche 3710, Inria, 1999. http://www.inria.fr/rrrt/rr-3710.html.

  39. C. Lemaréchal and F. Oustry. Nonsmooth algorithms to solve semidefinite programs. In L. El Ghaoui and S-I. Niculescu, editors, Advances in Linear Matrix Inequality Methods in Control, Advances in Design and Control, 2, pages 57–77. SIAM, 2000.

    Google Scholar 

  40. C. Lemaréchal, F. Oustry, and C. Sagastizábal. The U-lagrangian of a convex function. Transactions of the AMS, 352(2):711–729, 2000.

    MathSciNet  MATH  Google Scholar 

  41. C. Lemaréchal, F. Pellegrino, A. Renaud, and C. Sagastizábal. Bundle methods applied to the unit-commitment problem. In J. Doležal and J. Fidler, editors, System Modelling and Optimization, pages 395–402, 1996.

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  44. C. Lemaréchal and J. Zowe. Some remarks on the construction of higher order algorithms in convex optimization. Applied Mathematics and Optimization, 10(1):51–68, 1983.

    MathSciNet  MATH  Google Scholar 

  45. C. Lemaréchal and J. Zowe. The eclipsing concept to approximate a multi-valued mapping. Optimization, 22(1):3–37, 1991.

    MathSciNet  MATH  Google Scholar 

  46. L. Lovász. On the Shannon capacity of a graph. IEEE Transactions on Information Theory, IT 25:1–7, 1979.

    MathSciNet  MATH  Google Scholar 

  47. D.G. Luenberger. Optimization by Vector Space Methods. Wiley, New York, 1969.

    MATH  Google Scholar 

  48. L. Luksan and J. Vlcek. A bundle-newton method for nonsmooth unconstrained minimization. Mathematical Programming, 83A(3):373–391, 1998.

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  50. R.E. Marsten, W.W. Hogan, and J.W. Blankenship. The boxstep method for largescale optimization. Operations Research, 23(3):389–405, 1975.

    MathSciNet  MATH  Google Scholar 

  51. R. Mifflin. A quasi-second-order proximal bundle algorithm. Mathematical Programming, 73(1):51–72, 1996.

    MathSciNet  MATH  Google Scholar 

  52. R. Mifflin and C. Sagastizábal. On VU-theory for functions with primal-dual gradient structure. SIAM Journal on Optimization, 11(2):547–571, 2000.

    MathSciNet  MATH  Google Scholar 

  53. R. Mifflin, D.F. Sun, and L.Q. Qi. Quasi-Newton bundle-type methods for nondifferentiable convex optimization. SIAM Journal on Optimization, 8(2):583–603, 1998.

    MathSciNet  MATH  Google Scholar 

  54. A.S. Nemirovskii and D. Yudin. Informational complexity and efficient methods for the solution of convex extremal problems. Ékonomika i Mathematicheskie Metody, 12:357–369, 1976. (in Russian. English translation: Matekon, 13, 3-25).

    MATH  Google Scholar 

  55. Yu.E. Nesterov. Complexity estimates of some cutting plane methods based on the analytic barrier. Mathematical Programming, 69(1):149–176, 1995.

    MathSciNet  MATH  Google Scholar 

  56. Yu.E. Nesterov and A.S. Nemirovskii. Interior-Point Polynomial Algorithms in Convex Programming. Number 13 in SIAM Studies in Applied Mathematics. SIAM, Philadelphia, 1994.

    MATH  Google Scholar 

  57. Yu.E. Nesterov and A.S. Nemirovskii. Interior-Point Polynomial Algorithms in Convex Programming. SIAM Studies in Applied Mathematics 13. SIAM, Philadelphia, 1994.

    MATH  Google Scholar 

  58. J. Nocedal and S.J. Wright. Numerical Optimization. Springer Verlag, New York, 1999.

    MATH  Google Scholar 

  59. F. Oustry. A second-order bundle method to minimize the maximum eigenvalue function. Mathematical Programming, 89(1):1–33, 2000.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  61. B.T. Polyak. A general method for solving extremum problems. Soviet Mathematics Doklady, 8:593–597, 1967.

    MATH  Google Scholar 

  62. B.N. Pshenichnyi. The Linearization Method for Constrained Optimization. Springer Verlag, 1994.

    Google Scholar 

  63. C.R. Reeves. Modern Heuristic Techniques for Combinatorial Problems. Blackwell Scientific Publications, New York, 1993.

    MATH  Google Scholar 

  64. N. Shor. Utilization of the operation of space dilatation in the minimization of convex functions. Cybernetics, 6(1):7–15, 1970.

    Google Scholar 

  65. N.Z. Shor. Minimization methods for non-differentiable functions. Springer Verlag, Berlin, 1985.

    MATH  Google Scholar 

  66. N.Z. Shor and A.S. Davydov. Method of opbtaining estimates in quadratic extremal probems with boolean variables. Cybernetics, 21(2):207–211, 1985.

    MATH  Google Scholar 

  67. N.Z. Shor and N.G. Zhurbenko. A method for minimization, using the spacedilation operation in the direction of difference between two gradient sequences. Cybernetics, 7(3):450–459, 1971.

    Google Scholar 

  68. V.N. Solov'ev. The subdifferential and the directional derivatives of the maximum of a family of convex functions. Izvestiya: Mathematics, 62(4):807–832, 1998.

    MathSciNet  MATH  Google Scholar 

  69. G. Sonnevend. An “analytical centre” for polyhedra and new classes of global algorithms for linear (smooth, convex) programming. In A. Prékopa, J. Szelezsan, and B. Strazicky, editors, Proc. 12th IFIP Conf. System Modelling and Optimization, L.N. in Control and Information Sciences, pages 866–875. Springer Verlag, Berlin, 1986.

    Google Scholar 

  70. D.M. Topkis. A note on cutting-plane methods without nested constraint sets. Operations Research, 18:1216–1224, 1970.

    MathSciNet  MATH  Google Scholar 

  71. T. Terlaky. On lp programming. European Journal of Operational Research, 22:70–100, 1985.

    MathSciNet  Google Scholar 

  72. H. Uzawa. Iterative methods for concave programming. In K. Arrow, L. Hurwicz, and H. Uzawa, editors, Studies in Linear and Nonlinear Programming, pages 154–165. Stanford University Press, 1959.

    Google Scholar 

  73. L. Vandenberghe and S. Boyd. Semidefinite programming. SIAM Review, 38(1):49–95, 1996.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  75. L.A. Wolsey. Integer Programming. Wiley-Interscience, 1998.

    Google Scholar 

  76. S.J. Wright. Primal-Dual Interior-Point Methods. SIAM Publication, Philadelphia, 1997.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Inria, 655 avenue de l’Europe, 38334, Montbonnot, Saint Ismier, France

    Claude Lemaréchal

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  1. Claude Lemaréchal
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Editors and Affiliations

  1. Universität zu Köln,Institut für Informatik, Pohligstr.1, 50969, Köln, Germany

    Michael Jünger

  2. ENSIMAG - antenne de Montbonnot Laboratoire Informatique et Distribution, ZIRST, 51,avenue Jean Kuntzmann, 38330, Montbonnot Saint Martin, France

    Denis Naddef

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Lemaréchal, C. (2001). Lagrangian Relaxation. In: Jünger, M., Naddef, D. (eds) Computational Combinatorial Optimization. Lecture Notes in Computer Science, vol 2241. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45586-8_4

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  • DOI: https://doi.org/10.1007/3-540-45586-8_4

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