# A generalized Dantzig—Wolfe decomposition principle for a class of nonconvex programming problems

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## Abstract

Since Dantzig—Wolfe's pioneering contribution, the decomposition approach using a pricing mechanism has been developed for a wide class of mathematical programs. For convex programs a linear space of Lagrangean multipliers is enough to define price functions. For general mathematical programs the price functions could be defined by using a subclass of nondecreasing functions. However the space of nondecreasing functions is no longer finite dimensional. In this paper we consider a specific nonconvex optimization problem min {*f*(*x*):*h*_{ j }(*x*)⩽*g*(*x*),*j*=1, ⋯,*m, x*∈*X*}, where*f*(·),*h*_{ j }(·) and*g*(·) are finite convex functions and*X* is a closed convex set. We generalize optimal price functions for this problem in such a way that the parameters of generalized price functions are defined in a finite dimensional space. Combining convex duality and a nonconvex duality we can develop a decomposition method to find a globally optimal solution.

## Key words

Decomposition price functions global optimization## Preview

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## References

- [1]E. Asplund, “Differentiability of the metric projection in finite dimensional Euclidean space,”
*Proceedings of the American Mathematical Society*38 (1973) 218–219.Google Scholar - [2]R.E. Burkard, H.W. Hamacher and J. Tind, “On general decomposition schemes in mathematical programming,”
*Mathematical Programming Study*24 (1985) 238–252.Google Scholar - [3]R.E. Burkard, H.W. Hamacher and J. Tind, “On abstract duality in mathematical programming,”
*Zeitschrift für Operations Research*26 (1982) 197–209.Google Scholar - [4]F.H. Clarke,
*Optimization and Nonsmooth Analysis*(Wiley, New York, 1983).Google Scholar - [5]G.B. Dantzig and P. Wolfe, “The decomposition algorithm for linear programs,”
*Econometrica*29 (1961) 767–778.Google Scholar - [6]A.M. Geoffrion, “Primal resource-directive approaches for optimizing nonlinear decomposable systems,” Working paper 141, the RANK Corporation (Santa Monica, CA, 1968).Google Scholar
- [7]R.J. Hillestad and S.E. Jacobsen, “Reverse convex programming,”
*Applied Mathematics and Optimization*6 (1980) 63–78.Google Scholar - [8]J.B. Hiriart-Urruty, “Generalized differentiability, duality and optimization for problems dealing with differences of convex functions,”
*Lecture Notes in Economics and Mathematical Systems No. 256*(Springer, Berlin, 1985) pp. 37–69.Google Scholar - [9]R. Horst and H. Tuy,
*Global Optimization: Deterministic Approaches*(Springer, Berlin, 1990).Google Scholar - [10]T. Kuno, Y. Yajima and H. Konno, “An outer approximation method for minimizing the product of several convex functions on a convex set,” IHSS Report N. 91-33, Institute of Human and Social Sciences, Tokyo Institute of Technology (Tokyo, 1991).Google Scholar
- [11]L.S. Lasdon,
*Optimization Theory for Large Systems*(Macmillan, New York, 1970).Google Scholar - [12]G.L. Nemhauser and Z. Ullmann, “A note on the generalized Lagrange multiplier solution to an integer programming problem,”
*Operations Research*16 (1968) 450–453.Google Scholar - [13]G.L. Nemhauser and L.A. Wolsey,
*Integer and Combinatorial Optimization*(Wiley, New York, 1988).Google Scholar - [14]R.T. Rockafellar,
*Convex Analysis*(Princeton University Press, Princeton, NY, 1970).Google Scholar - [15]P.T. Thach, “Quasiconjugates of functions, duality relationship between quasiconvex minimization under a reverse convex constraint and quasiconvex maximization under a convex constraint and its applications,”
*Journal of Mathematical Analysis and Applications*159 (1991) 299–322.Google Scholar - [16]P.T. Thach, R.E. Burkard and W. Oettli, “Mathematical programs with a two-dimensional reverse convex constraint,”
*Journal of Global Optimization*1 (1991) 145–154.Google Scholar - [17]P.T. Thach, “A decomposition method using a pricing mechanism for min concave cost flow problems with a hierarchical structure,”
*Mathematical Programming*53 (1992) 339–359.Google Scholar - [18]J. Tind and L.A. Wolsey, “An elementary survey of general duality theory in mathematical programming,”
*Mathematical Programming*21 (1981) 241–281.Google Scholar - [19]J. Tind, “Decomposition in global optimization,”
*Journal of Global Optimization*1 (1991) 131–144.Google Scholar - [20]H. Tuy, “A general deterministic approach to global optimization via d.c. programming,” in: J.-B. Hiriart-Urruty, ed.,
*Mathematics for Optimization*(Elsevier, Amsterdam, 1986) pp. 137–162.Google Scholar - [21]H. Tuy and N.V. Thuong, “Minimizing a convex function over the complement of a convex set,”
*Methods of Operations Research*49 (1985) 85–89.Google Scholar - [22]L.A. Wolsey, “Integer programming duality: price functions and sensitivity analysis,”
*Mathematical Programming*21 (1981) 241–281.Google Scholar