By Darwin Klingman (auth.), Prof. John M. Mulvey (eds.)
Read or Download Evaluating Mathematical Programming Techniques: Proceedings of a Conference Held at the National Bureau of Standards Boulder, Colorado January 5–6, 1981 PDF
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Extra resources for Evaluating Mathematical Programming Techniques: Proceedings of a Conference Held at the National Bureau of Standards Boulder, Colorado January 5–6, 1981
We consider the nonlinear program minimize f(x) = gO(x) (1) k=1,2, ... ,K subjecttogk(x)~O (2) where x is a real n-vector and all the functions gk(x) are twice differentiable real functions. Each of the functions gk(x) k O,l, ... (x) 1: J J je:Jk (3) where the c. are constant coefficients, the t. (x) are terms, which may J - J-- be constants or some functions of the variables. The index sets J k represent a partition of the set [1,2, ... ,M]. We assume that all the terms in the ",roblem are indexed consecutively starting with the first term in the objective function and ending with the last term in the K-th constraint.
Then we want the probability that Xo lies in the region selected by the choice of i l , i 2 , ... , im' given that the region is non-empty. that the region generated by Fo is non-empty. The reverse implication is argued similarly. An important corollary of the above analysis is that for a particular realized configuration of hyperplanes, each polytope so created is equally likely to be selected. This follows from choosing a representative interior point of each polytope and noting that each point is equally likely to lie in the polytope generated.
J. Burruss, J. Elam, and D. Klingman, "The Design of a Generator for Structured Network-Based Problems," Research Report, Center for Cybernetic Studies, The University of Texas at Austin, (1980). 6. A. Geoffrion, "Comments on Mathematical Programming Project Panel on Futures," SHARE XIV Meeting, Los Angeles, (1975). 7. F. Glover, D. Karney, and D. Klingman, "The Augmented Predecessor Index Method for Locating Stepping Stone Paths and Assigning Dual Prices in Distribution Problems," Transportation Science 6,2 (1972) 171-179.