Duality in Stochastic Linear and Dynamic Programming by Willem K. Klein Haneveld

By Willem K. Klein Haneveld

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It is quite conceivable, that one is willing to specify the maximal acceptable probability of shortage, whereas the evaluation of shortage in terms of costs causes difficulties, or the other way around. It is even possible that both specifications can be made, but that the results do not coincide. g. 1) when w gets its realization. We do not agree with this statement. g. 4) [23]. 3) - is far from easy in practice. But it must be said that similar difficulties arise in penalty models - except for recourse models with physical recourse actions which can be evaluated in monetary terms, but there have been only a few applications of this type.

Notice is covered (take U+ = {OJ). Similarly, if one likes to suppress the constraints x > 0, X+ can be defined as X. Maximization can, of course, be transformed into minimization, and < inequalities into> inequalities, but we will deal with both standard forms since it is convenient in the treatment of duality. >: V x X -+ JR has been singled out. ,x> is a linear form on V, for each x E X. A shorthand notation for the duality is . 1. A dual pair of linear programming problems. The- pair of linear programs (LP 1) (LP 2 ) minimizexEX{: L 1x > b, x > O} maximizeyEy{: L2y < c, Y > O} is called a dual pair if their data satisfy the following conditions: a.

SIAM Rev. 16, 309-339. 38. -B. WETS (19B3). Stochastic programming: solution techniques and approximation schemes. A. BACHEM, M. GROETSCHEL, B. ). Mathematical Programming: The State of the Art - Bonn 1982, Springer, Berlin-Heidelberg-New York-Tokyo, 566-603. 39. J. ZACKOVA (1966). On minimax solutions of stochastic linear programming problems. Casopis Pest. Math. 91, 423-429. 40. Numerical methods in stochastic programming, IIASA, Laxenburg; to be published. 41. Bibliography of stochastic programming; to be contained in [40].

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