By Eitan Altman

Beginning new instructions in examine in either discrete occasion dynamic platforms in addition to in stochastic regulate, this quantity makes a speciality of a large category of keep watch over and of optimization difficulties over sequences of integer numbers. this can be a counterpart of convex optimization within the environment of discrete optimization. the speculation constructed is utilized to the regulate of stochastic discrete-event dynamic platforms. a few purposes are admission, routing, carrier allocation and holiday keep watch over in queuing networks. natural and utilized mathematicians will take pleasure in examining the publication because it brings jointly many disciplines in arithmetic: combinatorics, stochastic techniques, stochastic keep watch over and optimization, discrete occasion dynamic structures, algebra.

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**Example text**

It is suﬃcient to show that any set Θ ⊂ AX sequentially normally compact at y¯ (with respect to the whole space Y ) is also SNC at y¯ with respect to the smaller Banach space AX . 22 for the surjective operator A: X → AX . 21 ensuring that codim AX < ∞ due to aﬀ Θ ⊂ AX . , there is a closed subspace Z ⊂ Y with AX Z = Y . Now denote by Nε (·; Θ| AX ) the set of ε-normals to Θ with respect to AX and take Θ arbitrary sequences yk → y¯, εk ↓ 0, and yk∗ ∈ Nεk (yk ; Θ| AX ) converging to ∗ zero in the weak topology of (AX )∗ .

We have yˆ∗ ∈ Y ∗ . Since ∇ f (¯ ∗ remains to prove that yˆ ∈ Nc2 ε (¯ y ; Θ) with some constant c2 > 0. To furnish this, we use again the metric regularity property for the mapping f and its strict derivative. 16) for f with some µ > 0, we ﬁnd x y ∈ f −1 (y) such that x y − x¯ ≤ µ y − y¯ . 16) for the operator ∇ f (¯ x ), we get xˆy ∈ ∇ f (¯ x )−1 (y − y¯) with 22 1 Generalized Diﬀerentiation in Banach Spaces x y − x¯ − xˆy = o( x y − x¯ ) . Now putting all the above together, one has lim sup Θ y →¯ y xˆ∗ , xˆy xˆ∗ , xˆy yˆ∗ , y − y¯ = lim sup ≤ lim sup max 0, −1 y − y¯ y − y¯ µ x y − x¯ Θ Θ y →¯ y y →¯ y = lim sup max 0, Θ y →¯ y ≤ µ lim sup max 0, ε + x f −1 (Θ) → ¯ x xˆ∗ , x y − x¯ µ−1 x y − x¯ x ∗ , x − x¯ x − x¯ ≤ 2µε .

The principal diﬀerence between tangential and normal approximations is that the former constructions provide local approximations of sets in primal spaces, while the latter ones are deﬁned in dual spaces carrying “dual” information for the study of local behavior. Being applied to epigraphs of extended-real-valued functions and graphs of set-valued mappings, tangential approximations generate corresponding directional derivatives/subderivatives of functions and graphical derivatives of mappings, while normal approximations relate to subdiﬀerentials and coderivatives, respectively; see below.