By Richard Bellman

ISBN-10: 0691079137

ISBN-13: 9780691079134

Currently there's no ebook description for this book.

Currently there's no e-book description for this book.

Currently there's no ebook description for this book.

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Extra resources for Applied Dynamic Programming

Sample text

40 2. 4 Let Ci ⊂ Rn , i = 1, . . , m, be nonempty closed sets. Supm pose that zi ∈ (Ci )∞ with i=1 zi = 0 implies that zi = 0 ∀i = 1, . . , m. 2 concerns a closedness criterion of the set of convex combinations of a ﬁnite number of nonempty closed sets Ci , i = 1, . . , m, of Rn . Deﬁne the following two sets: m S := x∈R |x= λi xi , λ ∈ ∆m , xi ∈ Ci , i = 1, . . 9) i=1 where λi ∗ xi := λi xi , xi , with xi ∈ Ci if λi > 0, with xi ∈ (Ci )∞ if λi = 0, and ∆m denotes the simplex in Rm . 5 For a ﬁnite collection of nonempty closed sets Ci ⊂ Rn , i = 1, .

A function, then for any x ∈ Rn , S(x) reduces to a singleton, and thus the above formulation coincides with the deﬁnition of continuous functions. The latter formulation also leads to the translation of upper semicontinuity in terms of inﬁnite ¯. Then, sequences. Let {xk } ∈ dom S be a convergent sequence with limit x the set-valued map is upper semicontinuous at x ¯ if x)) = 0, lim dist(yk , S(¯ k→∞ ∀ yk ∈ S(xk ). 2 A set-valued map S : Rn ⇒ Rm is lower semicontinuous (lsc) at x ¯ if for each open set N with N ∩ S(¯ x) = ∅, there exists a neighborhood U (¯ x) such that x ∈ U (¯ x) =⇒ S(x) ∩ N = ∅.

Then the following relations hold: ∗ . (a) dom σC ⊂ C∞ ∗ ∗ ⊂ dom σC . (b) If int C∞ = ∅, then int C∞ (c) If C is convex, then (dom σC )∗ = C∞ . ∗ . Then ∃ 0 = d ∈ C∞ such that d, y > 0. Since Proof. (a) Let y ∈ C∞ d ∈ C∞ , ∃tk → ∞, ∃xk ∈ C with t−1 k xk → d, and with d, y > 0, it follows that xk , y → +∞, proving that y ∈ dom σC . (b) Let y ∈ dom σC . Then ∃xk ∈ C with xk , y → +∞. Passing to subsequences if necessary we can assume without loss of generality that xk −1 xk → d = 0, with d ∈ C∞ , and hence xk −1 xk , y ≥ 0.