Non-smooth Analysis
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This chapter presents some basic mathematical theory from non-smooth analysis [10, 12, 37, 56, 77, 78, 147, 149]. The aim of this chapter is not to give a real introduction to non-smooth analysis as the above textbooks are much better suited for that task. Instead, the primary aim of the chapter is to make the reader is familiar with the terminology and notation used in this monograph. Moreover, it provides a compendium on non-smooth and convex analysis which is useful when reading the following chapters. The reader might want to look up how a mathematical term is exactly defined, making use of the index in combination with this chapter. We begin with a brief introduction to sets (Section 2.1). The notion of continuity of functions is relaxed in Section 2.2 to semi-continuity and the notion of the classical derivative of smooth functions is extended to generalised differentials for non-smooth functions in Section 2.3. Subsequently, we discuss set-valued functions in Section 2.4. Topics from convex analysis are reviewed in Section 2.5 and the subderivative is discussed in Section 2.6.
2.1 Sets A number of properties of sets and set-valued functions will be briefly reviewed. Let C be a subset of the normed space Rn , equipped with the Euclidean norm · . Definition 2.1 (Closed Set). A set C ⊂ Rn is closed if it contains all its limit points. Every limit point of a set C is the limit of some sequence {xk } with xk ∈ C for all k ∈ N. The boundary of a set C, denoted by bdry C, is the set of points which can be approached both from C and from the outside of C. We define the closure of a set C as the smallest closed set containing C, i.e. C = C ∪ bdry C. Furthermore, those points in C which are not on the boundary form the interior of C, int C = C\ bdry C. We can uniquely decompose the closure of a
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2 Non-smooth Analysis
set in its boundary and its interior: C = bdry C ∪ int C. A set is called open, if it does not contain any of its boundary points, i.e. C ∩ bdry C = ∅. It holds that int C is an open set. Definition 2.2 (Bounded Set). A set C ⊂ Rn is bounded if there exists a point y ∈ Rn and a finite number c > 0 such that x − y < c for all x ∈ C. A set is bounded if it is contained in a ball of finite radius. Definition 2.3 (Compact Set). A set C ⊂ Rn is compact if it is closed and bounded. An important property in Non-smooth Analysis is the convexity of sets. Definition 2.4 (Convex Set). A set C ⊂ Rn is convex if for each x ∈ C and y ∈ C also (1 − q)x + qy ∈ C for arbitrary q with 0 ≤ q ≤ 1. It follows that a convex set contains all line segments between any two points in the set. Definition 2.5 (Convex Hull). The convex hull of a set C ⊂ Rn , denoted by co(C), is the smallest convex set that contains all the points of C. The convex hull of a set C is therefore the intersection of all the convex sets containing C. Consequently, the closed convex hull of {x, y} ∈ Rn is the line segment between x and y, i.e. the smallest closed convex set containing x and y co{x, y} = {(1 − q)x + qy, ∀ q ∈ [0, 1]}. (2.1) Figure 2.1 illu
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