Linear Programming 2: Theory and Extensions by George B. Dantzig, Mukund N. Thapa

By George B. Dantzig, Mukund N. Thapa

There's an instance of piecewise linear convex challenge which I was once trying to find.

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SCHWEIGMAN (1985). Operations Research ProbZems in AgricuZture in DeveZoping Countries, to be published. 32. D. -B. WETS (1967). Stochastic programs with recourse. SIAM J. AppZ. Math. 15, 1299-1314. 33. J. WESSELS (1967). Stochastic programming. Statist. NeerZandica 21, 39-53. 34. -B. WETS (1966). Programming under uncertainty: the complete problem. Z. Wahrsch. Verw. Gebiete 4, 316-339. 35. -B. liETS (1970). Problemes duaux en programmation stochastique. R. Acad. Sci. Ser. A-B 270, A47-ASO. 47 36.

The vector of decisions at stage t is x t ; it has to be chosen in the set Ct. The constraints Ax = b have a lower block triangular structure; the t-th block of constraints represents the recourse at stage t, where Att is the recourse matrix. 7). 33). It is assumed that all coefficients, not only the elements of the vectors c t and b t and of the matrices Ast but also those required to define Ct , say c~, constitute a random vector w with a known probability distribution. This vector is partitioned as W (w 1 ,w Z, ...

9). 10) is similar. e. 1 to a dual pair of linear programs. 5. Advanced Duality Theorem. Let (LP 1 ,LP 2 ) be a dual pair of linear programs with separated dualities. a. 11) < inf LP 1 = sup LP 2 < ~, and the supremum is attained if it is finite. b. 12) 00 > sup LP 2 = inf LP 1 > -00, and the infimum is attained if it is finite. PROOF. 1. Both references in the proof of this theorem consider only separated dualities. (b) Follows by reversing signs. REMARK. The boundedness condition on the optimal value function is not necessary for normality or even stability.

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