By Dimitri Bertsekas

ISBN-10: 1886529264

ISBN-13: 9781886529267

The 1st of the 2 volumes of the major and most recent textbook at the far-ranging algorithmic methododogy of Dynamic Programming, which might be used for optimum regulate, Markovian choice difficulties, making plans and sequential determination making less than uncertainty, and discrete/combinatorial optimization. The therapy makes a speciality of uncomplicated unifying topics, and conceptual foundations. It illustrates the flexibility, strength, and generality of the tactic with many examples and purposes from engineering, operations examine, and different fields. It additionally addresses widely the sensible software of the technique, almost certainly by utilizing approximations, and offers an creation to the far-reaching method of Neuro-Dynamic Programming. the 1st quantity is orientated in the direction of modeling, conceptualization, and finite-horizon difficulties, but additionally contains a noticeable creation to endless horizon difficulties that's compatible for lecture room use. the second one quantity is orientated in the direction of mathematical research and computation, and treats endless horizon difficulties widely. The textual content includes many illustrations, worked-out examples, and routines.

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**Extra resources for Dynamic Programming & Optimal Control, Vol. I**

**Sample text**

Clearly, both the functionals, (13) and (14), are sublinear, and vI′ (c; e′0 ) ≤ ′ vII (c; e′0 ). 1 (see also Remark 1 after it) in [47]. Proposition 5. Given e′0 ∈ dom vI′ (c; ·) := {e′ ∈ E ′ : vI′ (c; e′ ) < +∞}, the following assertions are equivalent: ′ (c; e′0 ); (a) vI′ (c; e′0 ) = vII ′ (b) the functional vII (c; ·) is weakly lower semi-continuous at e′0 . Let us deﬁne H := AK; then H 0 = (A∗ )−1 (K 0 ). Remark 3. Note that if dom vI′ (c; ·) ⊆ A∗ C(Ω)∗+ , (15) then, for every e′0 ∈ dom vI′ (c; ·), vI′ (c; e′0 ) = vI (c; µ0 ) and ′ vII (c; e′0 ) = vII (c; µ0 ) / dom vI′ (c; ·), one has whenever µ0 ∈ (A∗ )−1 (e′0 ).

Theorem 4. 1]). A multifunction F : X → L is L-cyclic monotone if and only if Q0 (ϕF ) is nonempty. Theorem 5. 2]). Suppose F : X → L is L-cyclic monotone. Given a function u : Z = dom F → IR ∪ {+∞}, the following statements are equivalent: (a) u ∈ Q0 (ϕF ); (b) u is a restriction to Z of some L-convex function U : X → IR ∪ {+∞}, and F (z) ⊆ ∂L U (z) for all z ∈ Z. Abstract Convexity and the Monge–Kantorovich Duality 51 The next result extending a classical convex analysis theorem due to Rockafellar is an immediate consequence of Theorems 4 and 5.

Abstract Convexity and the Monge–Kantorovich Duality 37 The original program is to maximize the linear functional h, µ0 := h(ω) µ0 (dω) subject to constraints: h ∈ H, h(ω) ≤ c(ω) for all ω ∈ Ω. Ω The optimal value of this program will be denoted as vI (c; µ0 ). , µ ∈ C(Ω)∗+ ) and µ ∈ µ0 − H 0 , where H 0 stands for the conjugate (polar) cone in C(Ω)∗+ , H 0 := {µ ∈ C(Ω)∗ : h, µ ≤ 0 for all h ∈ H}. The optimal value of this program will be denoted as vII (c; µ0 ). Thus, for any µ0 ∈ C(Ω)∗+ , one has vI (c; µ0 ) = sup{ h, µ0 : h ∈ H(c)}, (1) vII (c; µ0 ) = inf{c(µ) : µ ≥ 0, µ ∈ µ0 − H 0 }.

### Dynamic Programming & Optimal Control, Vol. I by Dimitri Bertsekas

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