By Peter Kall
I higghly suggest this booklet for a place to begin in Stochastic optimization. it truly is completely appropriate for an individual without historical past during this box. the simplest half is Wiley has published the copyright at the e-book, and the authors made the textual content on hand to the clinical group: it may be downloaded at no cost. Folllow the hyperlink on the Stochastic Programming group domestic web page (SP resources-Textbooks).
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Extra resources for Stochastic Programming (Wiley Interscience Series in Systems and Optimization)
Example text
1) By this assumption, we enforce the event {ξ ∈ Ξ | ψ(ξ) ∈ Bl − intBl } to have the natural measure µ({ξ ∈ Ξ | ψ(ξ) ∈ Bl − intBl }) = 0, which need not be true in general, as illustrated in Figure 13. 6) with nonempty interiors, they may be represented by inequality systems C l z ≤ 0, where C l = 0 is an appropriate matrix with no row equal to zero. 1), h(ξ) − T (ξ)x ∈ intBl . e. for any fixed j there exists a τˆlj > 0 such that C l [h(ξ) − T (ξ)(x ± τlj ej )] ≤ 0 34 Figure 12 STOCHASTIC PROGRAMMING Linear affine mapping of a polyhedron.
Ax ≥ b, where the decision variables are not required to be nonnegative—so-called free variables. In this case we may introduce a vector y ∈ IRm + of slack variables and—observing that any real number may be presented as the difference of two nonnegative numbers—replace the original decision vector x by the difference 6 We use occasionally “LP” as abbreviation for “linear program(ming)”. t. 1). Furthermore, it is easily seen that this transformed LP and its original formulation are equivalent in the sense that • given any solution (ˆ z + , zˆ− , yˆ) of the transformed LP, x ˆ := zˆ+ − zˆ− is a solution of the original version, • given any solution x ˇ of the original LP, the vectors yˇ := Aˇ x − b and zˇ+ , zˇ− ∈ IRn+ , chosen such that zˇ+ − zˇ− = x ˇ, solve the transformed version, and the optimal values of both versions of the LP coincide.
5 If g(·, ·) is jointly convex in (x, ξ) and P is quasi-concave, then the feasible set B(α) = {x|P ({ξ|g(x, ξ) ≤ 0}) ≥ α} is convex ∀α ∈ [0, 1]. 23), in general. 23), Fξ˜ being the distribution function of ξ, P ({ξ | T x ≥ ξ}) = Fξ˜(T x) ≥ α. e. if Fξ˜(λξ 1 + (1 − λ)ξ 2 ) ≥ min[Fξ˜(ξ 1 ), Fξ˜(ξ 2 )] for any two ξ 1 , ξ 2 ∈ Ξ and ∀λ ∈ [0, 1]. ✷ It seems worthwile to mention the following facts. If the probability measure P is quasi-concave then the corresponding distribution function Fξ˜ is quasiconcave.