By Claude Brezinski

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**Sample text**

Then for any ii with ATii 0 follows < < On the other hand, assume that ATu 0 always implies bTu 5 0. For an arbitrary 6 # 0 define c := AT&. Then Prop. 7 implies that the LP max{bTu I ATu 5 c ) is solvable. By the strong duality theorem, Prop. 1 1, 22 STOCHASTIC LINEAR PROGRAMMING its dual, min{cTx feasible. 5) is associated with a dual feasible basic solution. 3) has been derived, which satisjies the simplex criterion dT = c{B}~B-'N - cIN) 5 0. e. 8), is associated. ( Proofi Using the basis B, the matrix of the primal LP can be rewritten as A = (B, N).

Our assumption implies the inequality for arbitary x,. 28) of LPDesc (n,xhn) and LPMast (n,xh, 1, respectively. 20 fi), LPDesc (n,xh,) is also infeasible and consequently Fn(xhn) = + m holds. Thus we have Fn(xhn)= Fn(xhn) = + m . 28) is feasible. 28), at least one feasibility-cut constraint in the latter is violated. 38) is +m. 28) results in our inequality. 48 STOCHASTIC LINEAR PROGRAMMING (ii): By part (i) of this proposition, our assumption implies that F~(x,)5 Fm(xn)holds for all m E C(n)and arbitrary x,.

8). 8) into the standard form, we want to solve the problem 20 STOCHASTIC LINEAR PROGRAMMING To this LP we assign analogously the dual LP s. t. t. 1) again. Hence, the dual of the dual LP is the primal program again and we therefore can speak of a pair of dual LP's. There are further relations between the primal and the dual LP which are less obvious. First, we have the weak duality theorem. 8), respectively, it holds that bTii 5 cT5. Pro08 According to the assumed feasibilities A5 = b, 5 2 0, and ATii 5 c it follows that bTii = ( ~ 5 ) ~=i~i ~ ( ~ 5 ZTc.