By Darwin Klingman (auth.), Prof. John M. Mulvey (eds.)
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Additional info for Evaluating Mathematical Programming Techniques: Proceedings of a Conference Held at the National Bureau of Standards Boulder, Colorado January 5–6, 1981
We shall assume for simplicity that the program is to have two (local) miniQa at the points x* and y*, with the corresponding Lagrange multipliers u* and v* (u* and v* are Kvectors). The results apply to any number of specified solution points. Under our assumptions, the Lagrangian function k L(x,u) = f(x) - ~ k=l ukgk(x) (4 ) has to satisfy °xL(x*,u*) 0 0yL (y* ,v*) 0 ukg k (x*) 0 u* k (y*) 0 v* -> k v~gk > :} (5 ) k 1,2, ... x. In addition to the necessary conditions (5) and (6), the 90ints x* and y* must also be feasible, na,nely, satisfy the constraints (2).
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. 8. F. Glover, D. Karney, and D. Klingman, "Implementation and Computational Study on Start Procedures and Basis Change Criteria for a Primal Network Code," Networks 4,3 (1974) 191-212.
1. Introduction We consider in this paper the probleM of generating and character·· izing random feasible regions of linearly constrained mathematical programs. In particular, a large class of such random feasible regions (or convex polytopes) is operationally defined, and efficient Monte Carlo procedures for their generation are discussed. "lave taken on ne~T importance. Many of these evaluation schemes rely on the results of applying the algorithns to a series of randomly generated test problems.