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A primal-dual conjugate subgradient algorithm for specially structured linear and convex programming problems

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Abstract

This paper presents a primal-dual conjugate subgradient algorithm for solving convex programming problems. The motivation, however, is to employ it for solving specially structured or decomposable linear programming problems. The algorithm coordinates a primal penalty function and a Lagrangian dual function, in order to generate a (geometrically) convergent sequence of primal and dual iterates. Several refinements are discussed to improve the performance of the algorithm. These are tested on some network problems, with side constraints and variables, faced by the Freight Equipment Management Program of the Association of American Railroads, and suggestions are made for implementation.

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References

  1. Adams W and Sherali HD (1986) A Tight Linearization and an Algorithm for Zero-One Quadratic Programming Problems. Management Science 32(10):1274–1290

    Google Scholar 

  2. Agmon S (1954) The Relaxation Method for Linear Inequalities. Canadian Journal of Mathematics 6:382–392

    Google Scholar 

  3. Bartels R (1980) A Penalty Linear Programming Method Using Reduced-Gradient Basis-Exchange Techniques. Linear Algebra and Its Applications 29:17–32

    Google Scholar 

  4. Bazaraa MS and Sherali HD (1981) On the Choice of Step Size in Subgradient Optimization. European Journal of Operational Research 7:380–388

    Google Scholar 

  5. Bazaraa MS and Shetty CM (1979) Nonlinear Programming: Theory and Algorithms. Wiley, New York

    Google Scholar 

  6. Bertsekas DP (1976) Multiplier Methods: A Survey. Automatica 12:133–145

    Google Scholar 

  7. Bertsekas DP and Tseng P (1985) Relaxation Methods for Minimum Cost Ordinary and Generalized Network Flow Problems. LIDS Report P-1462, MIT

  8. Bitran G and Hax A (1976) On the Solution of Convex Knapsack Problems With Bounded Variables. Proceedings of the IXth International Symposium on Mathematical Programming, Budapest, pp 357–367

  9. Buys JD (1972) Dual Algorithms for Constrained Optimization Problems. Unpublished Ph.D. Thesis, University of Leiden, The Netherlands

    Google Scholar 

  10. Camerini PM, Fratta F and Maffioli F (1975) On Improving Relaxation Methods by Modified Gradient Techniques. Mathematical Programming Study 3:26–34

    Google Scholar 

  11. Cohen G and Zhu DL (1984) Decomposition Coordination Methods in Large-Scale Optimization Problems: The nondifferentiable Case and the Use of Augmented Lagrangians. In: JB Cruz (ed) Advances in Large-Scale Systems, vol 1. JAI Press, Greenwich, CT, pp 203–266

    Google Scholar 

  12. Conn AR (1976) Linear Programming Via a Nondifferentiable Penalty Function. SIAM Journal of Numerical Analysis 13(1):145–154

    Google Scholar 

  13. Dantzig GB and Wolfe P (1960) Decomposition Principle for Linear Programs. Operations Research 8:101–111

    Google Scholar 

  14. Etcheberry J (1977) The Set Covering Problem: A New Implicit Enumeration Algorithm. Operations Research 25:760–772

    Google Scholar 

  15. Fisher ML (1981) Lagrangian Relaxation Methods for Combinatorial Optimization. Management Science 27:1–18

    Google Scholar 

  16. Fisher ML and Jaikumar R (1981) A Generalized Assignment Heuristic for Vehicle Routing. Networks 11:109–124

    Google Scholar 

  17. Fletcher R (1983) Penalty Functions. In: A Bachem, M Grotschel and B Korte (eds) Mathematical Programming: The State of the Art. Springer-Verlag, Berlin, pp. 87–114

    Google Scholar 

  18. Fukushima M (1984) A Descent Algorithm for Nonsmooth Convex Optimization. Mathematical Programming 30(2):163–175

    Google Scholar 

  19. Gill PE, Murray W and Wright MH (1981) Practical Optimization. Academic Press, New York

    Google Scholar 

  20. Glickman TS and Sherali HD (1985) Large-Scale Network Distribution of Pooled Empty Freight Cars Over Time, with Limited Substitution and Equitable Benefits. Transportation Research B 19(2):85–94

    Google Scholar 

  21. Glover F and Klingman D (1981) The Simplex SON Algorithm for LP/Embedded Network Problems. Mathematical Programming Study 15:148–176

    Google Scholar 

  22. Golshtein EG (1981) An Iterative Linear Programming Algorithm Based on an Augmented Lagrangian. In: OL Mangasarian, RR Meyer and SM Robinson (eds) Nonlinear Programming, vol 4. Academic Press, New York, pp 131–146

    Google Scholar 

  23. Held M, Wolfe P and Crowder HP (1974) Validation of Subgradient Optimization. Mathematical Programming 6:62–88

    Google Scholar 

  24. Jeroslow RG and Lowe JK (1984) Modelling with Integer Variables. Mathematical Programming Study 22:167–184

    Google Scholar 

  25. Jeroslow RG and Lowe JK (1985) Experimental Results on the New Techniques for Integer Programming Formulations. Journal of the Operational Research Society 36:393–403

    Google Scholar 

  26. Kelley JE (1960) The Cutting Plane Method for Solving Convex Programs. SIAM Journal of Industrial and Applied Mathematics 8:703–712

    Google Scholar 

  27. Kiwiel K (1983) An Aggregate Subgradient Method for Nonsmooth Convex Minimization. Mathematical Programming 27:320–341

    Google Scholar 

  28. Korpelevich GM (1977) The Extragradient Method for Finding Saddle Points and Other Problems. Matekon 13(4):35–49

    Google Scholar 

  29. Lasdon L (1979) Optimization Theory for Large Systems. Macmillan, New York

    Google Scholar 

  30. Lemarechal C (1978) Bundle Methods in Nonsmooth Optimization. In: C Lemarechal and R Mifflin (eds) Nonsmooth Optimization. Pergamon Press, Oxford

    Google Scholar 

  31. Luenberger DG (1984) Linear and Nonlinear Programming, 2nd edn. Addision-Wesley, Reading, MA

    Google Scholar 

  32. Motzkin T and Schoenberg IJ (1954) The Relaxation Method for Linear Inequalities. Canadian Journal of Mathematics 6:393–404

    Google Scholar 

  33. Pierre DA and Lowe MJ (1975) Mathematical Programming via Augmented Lagrangians: An Introduction with Computer Programs. Addison-Wesley, Reading, MA

    Google Scholar 

  34. Poljak BT (1967) A General Method of Solving Extremum Problems. Soviet Mathematics Doklady 8(3):593–597

    Google Scholar 

  35. Poljak BT (1969) Minimization of Unsmooth Functionals. USSR Computational Mathematics and Mathematical Physics 9:14–29

    Google Scholar 

  36. Rockafellar RT (1973a) A Dual Approach to Solving Nonlinear Programming Problems by Unconstrained Optimization. Mathematical Programming 5:354–373

    Google Scholar 

  37. Rockafellar RT (1973b) The Multiplier Method of Hestenes and Powell Applied to Convex Programming. Journal of Optimization Theory and Applications 12:555–562

    Google Scholar 

  38. Rockafellar RT (1974) Augmented Lagrangian Multiplier Functions and Duality in Nonconvex Programming. SIAM Journal on Control and Optimization 12:268–285

    Google Scholar 

  39. Sen S and Sherali HD (1986) A Class of Convergent Primal-Dual Subgradient Algorithms for Decomposable Convex Programs. Mathematical Programming 35(3):279–297

    Google Scholar 

  40. Shah BV, Buehler RJ and Kempthorne O (1964) Some Algorithms for Minimizing a Function of Several Variables. Journals of the Society of Industrial and Applied Mathematics 12(1):74–91

    Google Scholar 

  41. Sherali HD and Myers DC (1988) Dual Formulations and Subgradient Optimization Strategies for Linear Programming Relaxations of Mixed-Integer Programs. Discrete Applied Mathematics 20:51–68

    Google Scholar 

  42. Sherali HD and Ulular O (1988) Conjugate Gradient Methods Using Quasi-Newton Updates with Inexact Line Searches. Journal of Mathematical Analysis and Applications (to appear)

  43. Shor NZ (1983) Generalized Gradient Methods of Nondifferentiable Optimization Employing Space Dilation Operators. In: A Bachem, M Grotschel and B Korte (eds) Mathematical Programming: The State of the Art. Springer-Verlag, Berlin, pp. 501–529

    Google Scholar 

  44. Stewart WR and Golden BL (1984) A Lagrangian Relaxation Heuristic for Vehicle Routing. European Journal of Operational Research 15:84–88

    Google Scholar 

  45. Tseng P and Bertsekas DP (1987) Relaxation Methods for Linear Programs. Mathematics of Operations Research 12(4):569–596

    Google Scholar 

  46. Ulular O (1988) A Primal-Dual Conjugate Subgradient Algorithm for Large-Scale/Specially Structured Linear Programming Problems. Unpublished Ph.D. Dissertation, Department of Industrial Engineering and Operations Research, Virginia Polytechnic Institute and State University, Blacksburg, VA

    Google Scholar 

  47. Van Roy TJ (1983) Cross Decomposition for Mixed Integer Programming. Mathematical Programming 25(1):46–63

    Google Scholar 

  48. Wolfe P (1975) A Method of Conjugate Subgradients for Minimizing Nondifferentiable Functions. Mathematical Programming Study 3:145–173

    Google Scholar 

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Authors and Affiliations

  1. Department of Industrial Engineering and Operations Research, Virginia Polytechnic Institute and State University, 24061, Blacksburg, VA, USA

    Hanif D. Sherali

  2. AT&T, 07981, Whippany, NJ, USA

    Osman Ulular

Authors
  1. Hanif D. Sherali
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  2. Osman Ulular
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Additional information

This research was supported by the Association of American Railroads.

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Sherali, H.D., Ulular, O. A primal-dual conjugate subgradient algorithm for specially structured linear and convex programming problems. Appl Math Optim 20, 193–221 (1989). https://doi.org/10.1007/BF01447654

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