Stochastic Programming - Index of

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RECOURSE PROBLEMS 223 In addition, we need to update the incumbent cut i k . This is done just the way we found cut k. Wesolve max{π T [ξ j − T (ξ j )x k−1 ] | π ∈ V } to obtain π j , and replace the old cut i k by θ ≥ 1 k k∑ (π j ) T [ξ j − T (ξ j )x] =α k i k−1 +(βi k k−1 ) T x. j=1 The remaining cuts are updated as before by letting θ ≥ k − 1 k [αk−1 j +(βj k−1 ) T x]+ 1 k Q = αk j +(βk j )T x for j =1,...,k−1, j≠ i k−1 . Now, it is time to check if the incumbent should be changed. We shall use Figure 31 for illustration, and we shall use the function φ k (x) defined earlier. In the figure we have k = 3. When we entered iteration k, our approximation of φ(x) was given by φ k−1 (x). Our incumbent solution was x k−1 and our iterate was x k . We show this in the top part of Figure 31 as x 2 and x 3 . The position of x 2 is somewhat arbitrary, since we cannot know how things looked in the previous iteration. Therefore φ k−1 (x k )−φ k−1 (x k−1 ) ≤ 0 was our approximation of how much we should gain by making x k our new incumbent. However, x k mightbeinanareawhereφ k−1 (x) is a bad approximation of φ(x). The function φ k (x), on the other hand, was developed around x k ,and should therefore be good in that area (in addition to being approximately as good as φ k−1 (x) around x k−1 ). This can be seen in the bottom part of Figure 31, where φ 3 (x) is given. The function φ 3 (x) is based on three cuts. One is new, the other two are updates of the two cuts in the top part of the figure, according to (8.1). Hence φ k (x k ) − φ k (x k−1 ) (a negative number) is a measure of how much we actually gained. If φ k (x k ) − φ k (x k−1 )
224 STOCHASTIC PROGRAMMING Figure 31 Calculations to find out if the incumbent should be changed.
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Stochastic Programming Second Editi
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Contents Preface . . . . . . . . .
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CONTENTS v 3.6 Simple Recourse . .
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Preface Over the last few years, bo
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PREFACE ix more explicitly with the
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1 Basic Concepts 1.1 Motivation By
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BASIC CONCEPTS 3 solutions. These a
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BASIC CONCEPTS 5 will be lost. In s
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BASIC CONCEPTS 7 1.2 Preliminaries
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BASIC CONCEPTS 9 with center ˆx an
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BASIC CONCEPTS 11 Figure 2 Determin
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BASIC CONCEPTS 13 (except for U). S
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BASIC CONCEPTS 15 may be wait-and-s
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BASIC CONCEPTS 17 dual decompositio
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BASIC CONCEPTS 19 and an empirical
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BASIC CONCEPTS 21 1.4 Stochastic Pr
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BASIC CONCEPTS 23 Figure 8 Measure
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BASIC CONCEPTS 25 These properties
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BASIC CONCEPTS 27 Figure 10 Classif
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BASIC CONCEPTS 29 Figure 12 Integra
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BASIC CONCEPTS 31 µ(A) =0alsoP (A)
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BASIC CONCEPTS 33 Hence, taking int
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BASIC CONCEPTS 35 Consequently, for
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BASIC CONCEPTS 37 Proof For ˆx, ¯
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BASIC CONCEPTS 39 Figure 13 Linear
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BASIC CONCEPTS 41 Figure 15 Differe
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BASIC CONCEPTS 43 However—for fin
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BASIC CONCEPTS 45 Figure 17 Induced
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BASIC CONCEPTS 47 Hence the feasibl
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BASIC CONCEPTS 49 Figure 19 λ = 1
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BASIC CONCEPTS 51 and hence P (λS
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BASIC CONCEPTS 53 The sequence of s
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BASIC CONCEPTS 55 is satisfied. Giv
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BASIC CONCEPTS 57 Now either z is a
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BASIC CONCEPTS 59 Figure 22 Polyhed
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BASIC CONCEPTS 61 Figure 23 Polyhed
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BASIC CONCEPTS 63 Figure 25 LP: unb
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BASIC CONCEPTS 65 this case, the re
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BASIC CONCEPTS 67 Step 3 Exchange t
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BASIC CONCEPTS 69 bases for any lin
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BASIC CONCEPTS 71 ✷ Example 1.6 C
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BASIC CONCEPTS 73 which, observing
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BASIC CONCEPTS 75 is equal to the p
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BASIC CONCEPTS 77 solve the program
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BASIC CONCEPTS 79 {v | Wv =0,q T v0
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BASIC CONCEPTS 81 The feasible set
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BASIC CONCEPTS 83 1.8.1 The Kuhn-Tu
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BASIC CONCEPTS 85 Figure 28 Kuhn-Tu
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BASIC CONCEPTS 87 Figure 29 The Sla
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BASIC CONCEPTS 89 and observing tha
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BASIC CONCEPTS 91 where the bounded
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BASIC CONCEPTS 93 λŷ +(1− λ)ˆ
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BASIC CONCEPTS 95 “reasonable”
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BASIC CONCEPTS 97 1.8.2.3 Penalty m
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BASIC CONCEPTS 99 To simplify the d
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BASIC CONCEPTS 101 Now let us come
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BASIC CONCEPTS 103 Wait-and-see pro
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BASIC CONCEPTS 105 y ≤ e −x } f
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BASIC CONCEPTS 107 Optimization: No
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2 Dynamic Systems 2.1 The Bellman P
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112 STOCHASTIC PROGRAMMING purpose
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114 STOCHASTIC PROGRAMMING In this
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116 STOCHASTIC PROGRAMMING Proof Th
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118 STOCHASTIC PROGRAMMING A 10% fe
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120 STOCHASTIC PROGRAMMING Stage 0
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124 STOCHASTIC PROGRAMMING Table 1
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128 STOCHASTIC PROGRAMMING situatio
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130 STOCHASTIC PROGRAMMING 2.5 Stoc
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132 STOCHASTIC PROGRAMMING Using th
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134 STOCHASTIC PROGRAMMING 2.6 Scen
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136 STOCHASTIC PROGRAMMING Today Fi
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138 STOCHASTIC PROGRAMMING procedur
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140 STOCHASTIC PROGRAMMING the natu
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142 STOCHASTIC PROGRAMMING book, be
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144 STOCHASTIC PROGRAMMING • A tw
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146 STOCHASTIC PROGRAMMING Figu
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148 STOCHASTIC PROGRAMMING 2.8.1 A
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150 STOCHASTIC PROGRAMMING 2.8.2 Fu
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152 STOCHASTIC PROGRAMMING 2.9.2 De
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154 STOCHASTIC PROGRAMMING between
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156 STOCHASTIC PROGRAMMING an exerc
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158 STOCHASTIC PROGRAMMING
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162 STOCHASTIC PROGRAMMING Hξ pos
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164 STOCHASTIC PROGRAMMING pos W po
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166 STOCHASTIC PROGRAMMING Figure
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274 STOCHASTIC PROGRAMMING directio
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276 STOCHASTIC PROGRAMMING [5] Gree
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278 STOCHASTIC PROGRAMMING outlined
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280 STOCHASTIC PROGRAMMING since we
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282 STOCHASTIC PROGRAMMING 2 1 a b
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286 STOCHASTIC PROGRAMMING 6.2.1 Th
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288 STOCHASTIC PROGRAMMING a soluti
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290 STOCHASTIC PROGRAMMING Since th
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292 STOCHASTIC PROGRAMMING It is no
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294 STOCHASTIC PROGRAMMING [-1,1] [
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296 STOCHASTIC PROGRAMMING for some
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298 STOCHASTIC PROGRAMMING +/- 1 1
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300 STOCHASTIC PROGRAMMING 1 (-1,0)
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302 STOCHASTIC PROGRAMMING with a g
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304 STOCHASTIC PROGRAMMING First, i
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306 STOCHASTIC PROGRAMMING Table 1
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308 STOCHASTIC PROGRAMMING Figure 2
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310 STOCHASTIC PROGRAMMING Universi
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314 STOCHASTIC PROGRAMMING duality
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316 STOCHASTIC PROGRAMMING best-so-
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