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# Download Martingales and Markov chains: solved exercises and theory by Paolo Baldi PDF By Paolo Baldi

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Extra info for Martingales and Markov chains: solved exercises and theory

Sample text

T (ω)x + W (ω)y = h(ω), γj (ω) + qj (ω)T y ≤ v, dk (ω) y ≤ rk (ω), T j = 1, . . , J2 , k = 1, . . , K2 . 2) have to be redefined in an appropriate way. In order to avoid all these manipulations and unnecessary notational complications that come with such a conversion, we shall address polyhedral problems in a more abstract way. This will also help us to deal with multistage problems and general convex problems. 45): L(y, π; x, ω) := f2 (y, ω) + π T h(ω) − T (ω)x − W (ω)y . We have inf L(y, π ; x, ω) = π T h(ω) − T (ω)x + inf f2 (y, ω) − π T W (ω)y y y = π h(ω) − T (ω)x − f2∗ (W (ω)T π, ω), T where f2∗ (·, ω) is the conjugate7 of f2 (·, ω).

45) can be written as Max π T h(ω) − T (ω)x − f2∗ (W (ω)T π, ω) . 46). 46). 2. 7 Note that since f2 (·, ω) is polyhedral, so is f2∗ (·, ω). ✐ ✐ ✐ ✐ ✐ ✐ ✐ 44 SPbook 2009/8/20 page 44 ✐ Chapter 2. 14. Let ω ∈ be given and suppose that Q(·, ω) is finite in at least one point x. ¯ Then the function Q(·, ω) is polyhedral (and hence convex). Moreover, Q(·, ω) is subdifferentiable at every x at which the value Q(x, ω) is finite, and ∂Q(x, ω) = −T (ω)T D(x, ω). 47) Proof. Let us define the function ψ(π ) := f2∗ (W T π ).

22)). 6. 1 in the case when all demand has to be satisfied, by making additional orders of the missing parts. In this case, the cost of each additionally ordered part j is rj > cj . Formulate the problem as a linear two-stage stochastic programming problem. 7. 3 in the case when all demand has to be satisfied, by backlogging the excessive demand, if necessary. In this case, it costs bi to delay delivery of a unit of product i by one period. Additional orders of the missing parts can be made after the last demand DT becomes known.