Convergence and Application of a Decomposition Method Using Duality Bounds for Nonconvex Global Optimization

  • N.V. Thoai


The subject of this article is a class of global optimization problems, in which the variables can be divided into two groups such that, in each group, the functions involved have the same structure (e.g. linear, convex or concave, etc.). Based on the decomposition idea of Benders (Ref. 1), a corresponding master problem is defined on the space of one of the two groups of variables. The objective function of this master problem is in fact the optimal value function of a nonlinear parametric optimization problem. To solve the resulting master problem, a branch-and-bound scheme is proposed, in which the estimation of the lower bounds is performed by applying the well-known weak duality theorem in Lagrange duality. The results of this article concentrate on two subjects: investigating the convergence of the general algorithm and solving dual problems of some special classes of nonconvex optimization problems. Based on results in sensitivity and stability theory and in parametric optimization, conditions for the convergence are established by investigating the so-called dual properness property and the upper semicontinuity of the objective function of the master problem. The general algorithm is then discussed in detail for some nonconvex problems including concave minimization problems with a special structure, general quadratic problems, optimization problems on the efficient set, and linear multiplicative programming problems.

Global optimization nonconvex optimization decomposition nonlinear parametric optimization branch-and-bound schemes 


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  1. 1.
    BENDERS, J. F., Partitioning Procedures for Solving Mixed-Variables Programming Problems, Numerische Mathematik, Vol. 4, pp. 238-252, 1962.Google Scholar
  2. 2.
    ROSEN, J. B., and PARDALOS, P. M., Global Minimization of Large-Scale Constrained Concave Quadratic Problems by Separable Programming, Mathematical Programming, Vol. 34, pp. 163-174, 1986.Google Scholar
  3. 3.
    FLOUDAS, C. A., and AGGARWAL, A., A Decomposition Strategy for Global Optimum Search in the Pooling Problem, ORSA Journal on Computing, Vol. 2, pp. 225-235, 1990.Google Scholar
  4. 4.
    FLOUDAS, C. A., and VISWESWARAN, V., A Primal-Relaxed Dual Global Optimization Approach, Journal of Optimization Theory and Applications, Vol. 78, pp. 187-225, 1993.Google Scholar
  5. 5.
    HORST, R., and THOAI, N. V., Conical Algorithms for the Global Minimization of Linearly Constrained Decomposable Concave Minimization Problems, Journal of Optimization Theory and Applications, Vol. 74, pp. 469-486, 1992.Google Scholar
  6. 6.
    HORST, R., and THOAI, N. V., Constraint Decomposition Algorithms in Global Optimization, Journal of Global Optimization, Vol. 5, pp. 333-348, 1994.Google Scholar
  7. 7.
    HORST, R., and THOAI, N. V., Decomposition Approach for the Global Minimization of Biconcave Functions over Polytopes, Journal of Optimization Theory and Applications, Vol. 88, pp. 561-583, 1996.Google Scholar
  8. 8.
    MUU, L. D., and OETTLI, W., Method for Minimizing a Convex-Concave Function over a Convex Set, Journal of Optimization Theory and Applications, Vol. 70, pp. 377-384, 1991.Google Scholar
  9. 9.
    THOAI, N. V., A Global Optimization Approach for Solving the Convex Multiplicative Programming Problem, Journal of Global Optimization, Vol. 1, pp. 341-357, 1991.Google Scholar
  10. 10.
    THOAI, N. V., Canonical DC Programming Techniques for Solving a Convex Program with an Additional Constraint of Multiplicative Type, Computing, Vol. 50, pp. 241-253, 1993.Google Scholar
  11. 11.
    GEOFFRION, A. M., Duality in Nonlinear Programming: A Simplified Application-Oriented Development, SIAM Review, Vol. 13, pp. 1-37, 1971.Google Scholar
  12. 12.
    MANGASARIAN, O. L., Nonlinear Programming, Robert E. Krieger Publishing Company, Huntington, New York, 1979.Google Scholar
  13. 13.
    BEN-TAL, A., EIGER, G., and GERSHOVITZ, V., Global Minimization by Reducing the Duality Gap, Mathematical Programming, Vol. 63, pp. 193-212, 1994.Google Scholar
  14. 14.
    DÑR, M., and HORST, R., Lagrange Duality and Partitioning Techniques in Nonconvex Global Optimization, Journal of Optimization Theory and Applications, Vol. 95, pp. 347-369, 1997.Google Scholar
  15. 15.
    BANK, B., GUDDAT, J., KLATTE, D., KUMMER, B., and TAMMER, K., Nonlinear Parametric Optimization, Birkhäuser, Basel, Switzerland, 1983.Google Scholar
  16. 16.
    DANTZIG, G. B., FOLKMAN, J., and SHAPIRO, N., On the Continuity of the Minimum Set of a Continuous Function, Journal of Mathematical Analysis and Applications, Vol. 17, pp. 519-548, 1967.Google Scholar
  17. 17.
    FIACCO, A. V., Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press, New York, NY, 1983.Google Scholar
  18. 18.
    BERGE, C., Topological Spaces, Macmillan, New York, NY, 1963.Google Scholar
  19. 19.
    HOGAN, W. W., Point-to-Set Maps in Mathematical Programming, SIAM Review, Vol. 15, pp. 591-603, 1973.Google Scholar
  20. 20.
    HORST, R., and TUY, H., Global Optimization: Deterministic Approaches, 3rd Edition, Springer, Berlin, Germany, 1996.Google Scholar
  21. 21.
    HORST, R., PARDALOS, P. M., and THOAI, N. V., Introduction to Global Optimization, Kluwer, Dordrecht, Netherlands, 1995.Google Scholar
  22. 22.
    FALK, J. E., and HOFFMAN, K. L., A Successive Underestimation Method for Concave Minimization Problems, Mathematics of Operations Research, Vol. 1, pp. 251-259, 1976.Google Scholar
  23. 23.
    PHILIP, J., Algorithms for the Vector Maximization Problem, Mathematical Programming, Vol. 2, pp. 207-229, 1972.Google Scholar
  24. 24.
    BENSON, H. P., Optimization over the Efficient Set, Journal of Mathematical Analysis and Applications, Vol. 98, pp. 562-580, 1984.Google Scholar
  25. 25.
    DAUER, J. P., and FOSNAUGH, T. A., Optimization over the Efficient Set, Journal of Global Optimization, Vol. 7, pp. 261-277, 1995.Google Scholar
  26. 26.
    FÑLÖP, J., A Cutting Plane Algorithm for Linear Optimization over the Efficient Set, Generalized Convexity, Edited by S. Komlösi, T. Rapcsàk, and S. Schaible, Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, Vol. 405, pp. 374-385, 1994.Google Scholar
  27. 27.
    LE-THI, H. A., PHAM, D. T., and MUU, L. D., Numerical Solution for Optimization over the Efficient Set by DC Optimization Algorithms, Operations Research Letters, Vol. 19, pp. 117-128, 1996.Google Scholar
  28. 28.
    MUU, L. D., and LUC, L. T., On Equivalence between Convex Maximization and Optimization over the Efficient Set, Vietnam Journal of Mathematics, Vol. 24, pp. 439-444, 1996.Google Scholar
  29. 29.
    HORST, R., and THOAI, N. V., Maximizing a Concave Function over the Efficient or Weakly-Efficient Set, European Journal of Operations Research, Vol. 117, pp. 239-252, 1999.Google Scholar
  30. 30.
    THOAI, N. V., Conical Algorithm in Global Optimization for Optimizing over Efficient Sets, Journal of Global Optimization, Vol. 18, pp. 321-336, 2000.Google Scholar
  31. 31.
    KONNO, H., and KUNO, T., Linear Multiplicative Programming, Mathematical Programming, Vol. 56, pp. 51-64, 1992.Google Scholar
  32. 32.
    THOAI, N. V., Duality Bound Method for the General Quadratic Programming Problem with Quadratic Constraints, Journal of Optimization Theory and Applications, Vol. 107, pp. 331-354, 2000.Google Scholar

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© Plenum Publishing Corporation 2002

Authors and Affiliations

  • N.V. Thoai
    • 1
  1. 1.Department of MathematicsUniversity of TrierTrierGermany

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