By Gupta S. N.
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Additional info for A Chance Constrained Approach to Fractional Programming with Random Numerator
If f is directionally Lipschitzian at x, there exist y E E and Y E N(y) such that [O(x; 0) is bounded above on Y (see (20)). , (18)) implies that fL 00 (x; 0) (fU oo (x; 0)) is bounded above and> -00 on Y. Therefore assertion (a) is valid. 5 for B: = L, D: = U. 8. c. for each i E I(x o), and that (21) dom fLoo(X o: 0) () () int dom giUoo(Xo: 0) .. ~. I(xo) Then for each h E GUC(L, f, xo) and hi E GUC(U, gi' xo)' there exist Ai least one of which is nonzero, such that o E Aoi>h(O) + (22) (23) domfLOO(xO:o) () L 2 0, i E I(x o) U (O), at Aii>hi(O) 1(10) () (YlgiUOO(10:Y)<0)"~' 1(10 ) then Ao> 0 in (22).
If T is assumed to be P-convex. x)). x). Assuming that T is P-convex is less restrictive than the common approach of assuming that T is convex (see. for example. Banker et a1. (1984)). 4. paration Theorem An interesting theoretical and practical question is whether or not one would ever observe the "P-convex portion" of the technology. , those boundary points of the level set lying in the interior of the convex hull of the level set (points A. Band C in Figure 4). If the production planner were cost minimizing (one possible behavioral assumption), then points A.
A useful generalization of convexity must fulfill two requirements. First. and foremost, it must admit a rich class of economic models. Second. it must be tractable for analysis. e .. it 24 must have some useful mathematical structure. Recently, Hackman and Passy (1988) introduced a generalization of convexity called Projective-convexity or (P-convexity) according to the second approach. It was shown that P-convexity has the requisite mathematical structure. For example, P-convex sets possess a separation property (as in the first approach to generalizing convexity): a point not in a closed P-convex set can be separated by a quadrant, which is an intersection of closed halfspaces generated by orthogonal hyperplanes.
A Chance Constrained Approach to Fractional Programming with Random Numerator by Gupta S. N.