Abstract
This chapter considers an important generalization of the basic stochastic approximation scheme of Chapter 2, which we call ‘stochastic recursive inclusions’. The idea is to replace the map h: ℛd → ℛd in the recursion (2.1.1) of Chapter 2 by a set-valued map h: ℛd → {subsets of ℛd}, satisfying the following conditions:
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(i)
For each x ∈ ℛd, h(x) is convex and compact.
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(ii)
For all x ∈ ℛd,
$$\mathop {\sup }\limits_{y \in h\left( x \right)} \;\;\left\| y \right\| < K\left( {1 + \left\| x \right\|} \right)$$((5.1.1))for some K > 0.
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(iii)
h is upper semicontinuous in the sense that if x n → x and y n → y with y n ∈ h(x n ) for n ≥ 1, then y ∈ h(x). (In other words, the graph of h, defined as {(x, y): y { h(x)}, is closed.)
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© 2008 Hindustan Book Agency
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Borkar, V.S. (2008). Stochastic Recursive Inclusions. In: Stochastic Approximation. Texts and Readings in Mathematics, vol 48. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-38-5_5
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DOI: https://doi.org/10.1007/978-93-86279-38-5_5
Publisher Name: Hindustan Book Agency, Gurgaon
Print ISBN: 978-81-85931-85-2
Online ISBN: 978-93-86279-38-5
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