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

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  • N.V. Thoai

    (University of Trier)

Abstract

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.

Suggested Citation

  • N.V. Thoai, 2002. "Convergence and Application of a Decomposition Method Using Duality Bounds for Nonconvex Global Optimization," Journal of Optimization Theory and Applications, Springer, vol. 113(1), pages 165-193, April.
    • Handle: RePEc:spr:joptap:v:113:y:2002:i:1:d:10.1023_a:1014865432210
    DOI: 10.1023/A:1014865432210
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    References listed on IDEAS

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    1. James E. Falk & Karla R. Hoffman, 1976. "A Successive Underestimation Method for Concave Minimization Problems," Mathematics of Operations Research, INFORMS, vol. 1(3), pages 251-259, August.
    2. Christodoulos A. Floudas & Avanish Aggarwal, 1990. "A Decomposition Strategy for Global Optimum Search in the Pooling Problem," INFORMS Journal on Computing, INFORMS, vol. 2(3), pages 225-235, August.
    3. M. Dür & R. Horst, 1997. "Lagrange Duality and Partitioning Techniques in Nonconvex Global Optimization," Journal of Optimization Theory and Applications, Springer, vol. 95(2), pages 347-369, November.
    4. N. V. Thoai, 2000. "Duality Bound Method for the General Quadratic Programming Problem with Quadratic Constraints," Journal of Optimization Theory and Applications, Springer, vol. 107(2), pages 331-354, November.
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    Cited by:

    1. Le Thi, Hoai An & Pham, Dinh Tao & Thoai, Nguyen V., 2002. "Combination between global and local methods for solving an optimization problem over the efficient set," European Journal of Operational Research, Elsevier, vol. 142(2), pages 258-270, October.
    2. R. Horst & N. V. Thoai & Y. Yamamoto & D. Zenke, 2007. "On Optimization over the Efficient Set in Linear Multicriteria Programming," Journal of Optimization Theory and Applications, Springer, vol. 134(3), pages 433-443, September.
    3. Nguyen Thoai, 2012. "Criteria and dimension reduction of linear multiple criteria optimization problems," Journal of Global Optimization, Springer, vol. 52(3), pages 499-508, March.
    4. N. V. Thoai, 2010. "Reverse Convex Programming Approach in the Space of Extreme Criteria for Optimization over Efficient Sets," Journal of Optimization Theory and Applications, Springer, vol. 147(2), pages 263-277, November.
    5. Benson, Harold P., 2007. "A simplicial branch and bound duality-bounds algorithm for the linear sum-of-ratios problem," European Journal of Operational Research, Elsevier, vol. 182(2), pages 597-611, October.
    6. Sonia & Puri, M.C., 2008. "Two-stage time minimizing assignment problem," Omega, Elsevier, vol. 36(5), pages 730-740, October.

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