Convex Analysis and Minimization Algorithms I Fundamentals
Convex Analysis may be considered as a refinement of standard calculus, with equalities and approximations replaced by inequalities. As such, it can easily be integrated into a graduate study curriculum. Minimization algorithms, more specifically those ad
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Prerequisites. A good mastering ofthe following subjects: basic results from real analysis; definition and elementary properties of convex sets in il2 ; elementary geometry in the affine spaceJR.2• Introduction. Convex functions of a real variable form an important class of functions in the context of what is usually called real analysis. They are useful in optimization - as will be shown in this book - but also in several areas of applied mathematics, where their properties are often key ingredients to derive a priori bounds, sharp inequalities, etc. Even though general convex functions will be studied in extenso further on, there are several reasons to devote a special chapter to the one-dimensional case. (i) Convexity is essentially a one-dimensional concept, since it reduces to convexity on the line joining two arbitrary points x and x'. (ii) For theoretical as well as algorithmic purposes, the one-dimensional trace of a convex function f, i.e. the function t ~ f (x + t d) (t real), will have to be studied thoroughly anyway in later chapters. (iii) It is a good support for intuition; for example, the so-called subdifferential of a convex function can be introduced and studied very easily in the univariate case; we will also take this opportunity to introduce the concept of conjugacy operation, in this simplified setting. (iv) Some properties of convex functions are specific to one single variable; these properties, as well as many examples and counter-examples, will be included here. The material contained in this chapter provides, on the one hand, sufficient background for those readers wishing to know basic properties of one-dimensional convex functions, in order to apply them in other areas of applied mathematics. On the other hand, this chapter serves as an introduction to the rest ofthe book; most of its results will be proved rather quickly, since they will be proved subsequently in the multi-dimensional setting. The chapter can be skipped by a reader already familiar with properties of convex functions from the viewpoint of standard real analysis. We believe, however, that our presentation may be helpful for a better understanding of the whole book.
1 Basic Definitions and Examples The intervals fonn the simplest instances of subsets of R. We retain two among their possible definitions: a subset I C JR. is an interval if and only if, whenever x and x, belong to I, one of the following properties holds:
J.-B. Hiriart-Urruty et al., Convex Analysis and Minimization Algorithms I © Springer-Verlag Berlin Heidelberg 1993
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I. Convex Functions of One Real Variable
(i) every point between x and x' belongs to I (definition based on the natural ordering of JR.); (ii) for all a between 0 and 1, the point ax + (1 - a)x' belongs to I (definition using the vector structure ofR). The following classification of nonempty intervals is convenient: - the compact intervals: I = [a, b] (a, bE JR. with a ~ b); - the bounded but not closed intervals: [a, b[, la, b], la, b[ (a, b E JR., a < b); - the intervals majorized
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