Computational Experience of an Interior-Point SQP Algorithm in a Parallel Branch-and-Bound Framework

  • Eva K. Lee
  • John E. Mitchell
Part of the Applied Optimization book series (APOP, volume 33)


An interior-point algorithm within a parallel branch-and-bound framework for solving nonlinear mixed integer programs is described. The nonlinear programming relaxations at each node are solved using an interior point SQP method. In contrast to solving the relaxation to optimality at each tree node, the relaxation is only solved to near-optimality. Analogous to employing advanced bases in simplex-based linear MIP solvers, a “dynamic” collection of warmstart vectors is kept to provide “advanced warmstarts” at each branch-and-bound node. The code has the capability to run in both shared-memory and distributed-memory parallel environments. Preliminary computational results on various classes of linear mixed integer programs and quadratic portfolio problems are presented.


Interior Point Mixed Integer Sequential Quadratic Programming Interior Point Method Dual Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Applegate, D., R.E. Bixby, V. Chvátal, and W. Cook. “The traveling salesman problem,” DIMACS, Rutgers University, New Brunswick, NJ, 1994.Google Scholar
  2. [2]
    Bixby, R.E., W. Cook, A. Cox, and E.K. Lee. “Computational experience with parallel mixed integer programming in a distributed environment,” to appear in Annals of Operations Research, Special Issue on Parallel Optimization, Department of Computational and Applied Mathematics, Rice University, Houston, Texas, 1996.Google Scholar
  3. [3]
    Bixby, R.E., W. Cook, A. Cox, and E.K. Lee. “Parallel mixed integer programming,” Department of Computational and Applied Mathematics, Rice University, CRPC-TR95554, Houston, Texas, 1995Google Scholar
  4. [4]
    Bixby, R.E., and E.K. Lee. “Solving a truck dispatching scheduling problem using branch-and-cut,” Operations Research, Vol. 46, pp. 355–367, 1998.zbMATHCrossRefGoogle Scholar
  5. [5]
    Boggs, P.T., P.D. Domich, and J. E. Rogers. “An interior-point method for general large scale quadratic programming problems,” Annals of Operations Research, Vol. 62, pp. 419–438, 1996. Available at ftp: MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Boggs, P.T., J.W. Tolle, and A.J. Kearsley. “A practical algorithm for general large scale nonlinear optimization problems,” National Institute of Standards and Technology, Gaithersburg, MD 20899, April 1994. Available at Scholar
  7. [7]
    Borchers, B., and J.E. Mitchell. “Using an interior point method in a branch and bound algorithm for integer programming,” Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, Nr. 195, March, 1991. Revised July 7, 1992.Google Scholar
  8. [8]
    Borchers, B., and J.E. Mitchell. “An improved branch and bound algorithm for mixed integer nonlinear programming,” Computers and Operations Research, Vol. 21, No. 4, pp. 359–367, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Borchers, B., and J.E. Mitchell. “A computational comparison of Branch and Bound and Outer Approximation Methods for 0–1 mixed integer nonlinear programs,” Computers and Operations Research. Vol. 24, pp. 699–701, 1997.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Cannon, T.L., and K.L. Hoffman. “Large-scaled 0/1 linear programming on distributed workstations,” Annals of Operations Research, Vol. 22, pp. 181–217, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Domich, P.D., P.T. Boggs, J.E. Rogers, and C. Witzgall. “Optimizing over three-dimensional subspaces in an interior-point method for linear programming,” Linear Algebra and its Applications, Vol. 152, pp. 315–342, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Duran, M.A., and I.E. Grossman. “An outer-approximation algorithm for a class of mixed-integer nonlinear programs,” Mathematical Programming, Vol. 36, pp. 307–339, 1986.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Eckstein, J. “Parallel Branch-and-Bound Algorithms for General mixed Integer Programming on the CM-5,” SIAM Journal on Optimization, Vol. 4, pp. 794–81, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    El-Bakry, A.S., R.A. Tapia, and Y. Zhang. “A study of indicators for identifying zero variables in interior-point methods,” SIAM Review, Vol. 36, pp. 45–72, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    El-Bakry, A.S., R.A. Tapia, T. Tsuchiya, and Y. Zhang. “On the formulation and theory of the Newton interior-point method for nonlinear programming,” Department of Mathematical Sciences, Rice University, Houston, Texas 77251, December 1992.Google Scholar
  16. [16]
    Floudas, C.A. “Nonlinear and Mixed Integer Optimization,” Oxford University Press, 1995.zbMATHGoogle Scholar
  17. [17]
    Gendron, B., and T.G. Crainic. “Parallel branch-and-bound algorithms: survey and synthesis,” Operations Research, Vol. 42, pp. 1042–1066, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    George, J.A., and J.W.H. Liu. “Computer Solution of Large Sparse Positive Definite Systems,” Prentice-Hall, Englewood Cliffs, NJ, 1981.zbMATHGoogle Scholar
  19. [19]
    Hansen, P., B. Jaumard, and V. Mathon. “Constrained nonlinear 0–1 Programming,” ORSA Journal on Computing, Vol. 5, pp. 97–119, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Keleher, P., A. Cox, S. Dwarkadas, and W. Zwaenepoel. “Tread-Marks: Distributed memory on standard workstations and operating systems,” Proceedings of the 1994 Winter Usenix Conference, 115–131, 1994.Google Scholar
  21. [21]
    Lee, E.K. “Computational Experience of a General Purpose Mixed 0/1 Integer Programming Solver (MIPSOL),” Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA. Available at Scholar
  22. [22]
    Lee, E.K. “A Branch-and-Cut Approach to Treatment Plan Optimization for Permanent Prostate Implants,” Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, 1997. Available at Scholar
  23. [23]
    Li, K., and P. Hudak. “Memory coherence in shared virtual memory systems,” ACM Transactions on Computer Systems, Vol. 4, pp. 229–239, 1989.Google Scholar
  24. [24]
    Lustig, I.J., R.E. Marsten and D.F. Shanno. “On implementing Mehrotra’s predictor-corrector interior point method for linear programming,” SIAM Journal on Optimization, Vol. 2, pp. 435–449, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    Mauricio, D., and N. Maculan. “A Trust Region Method for Zero-One Nonlinear Programming,” RAIROOperations Research, Vol. 31, pp. 331–341, 1997.MathSciNetzbMATHGoogle Scholar
  26. [26]
    McCormick, G.P. “The superlinear convergence of a nonlinear primal-dual algorithm,” T-550/91, School of Engineering and Applied Science, George Washington University, Washington, D.C., 1991.Google Scholar
  27. [27]
    Michelon, P., and N. Maculan. “Lagrangean decomposition for integer nonlinear programming with linear constraints,” Mathematical Programming, Vol. 52, pp. 303–313, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    Mitchell, J.E. “Interior Point Algorithms for Integer Programming”, J. E. Beasley (eds.) Advances in Linear and Integer Programming, Ch. 6, 223–248, Oxford University Press, 1996.Google Scholar
  29. [29]
    Mitchell, J.E. “Interior Point Methods for Combinatorial Optimization”, Tamás Terlaky (eds.) Interior Point Methods in Mathematical Programming, Ch. 11, 417–466, Kluwer Academic Publishers, 1996. Scholar
  30. [30]
    Mitchell, J.E., and P.P. Pardalos and M.G.C. Resende. “Interior point methods for combinatorial optimization”, Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, November 1997. Accepted for publication in Handbook of Combinatorial Optimization, 1998. Scholar
  31. [31]
    Mitchell, J.E. “Computational experience with an interior point cutting plane algorithm”, Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180–3590, February 1997. Revised: April 1997. Scholar
  32. [32]
    Monteiro, R.D.C., and S.J. Wright. “A globally and superlinearly convergent potential reduction interior point method for convex programming”, SIE 92–13, SIE Department, University of Arizona, Tucson, AZ, 1992. Scholar
  33. [33]
    Murtagh, B.A., and M.A. Saunders. “(MINOS) 5.5 Users Guide”, SOL 83–20, 14, 1978. SOL, Stanford University, Palo Alto, CA, Revised: July 1998.Google Scholar
  34. [34]
    Nemhauser, G.L., and L.A. Wolsey. “Integer and Combinatorial Optimization”, John Wiley, New York, 1988.zbMATHGoogle Scholar
  35. [35]
    Saliinidis, N. “BARON: An all-purpose global optimization software package”, UILU-ENG-95–4002, Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, Illinois, 1995.Google Scholar
  36. [36]
    Vanderbei, R.J. “LOQO: An interior point code for quadratic programming”, Statistics and Operations Research, Princeton University, Princeton, NJ 08544, February 1995.Google Scholar

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© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Eva K. Lee
  • John E. Mitchell

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