Convex Set - Convex Sets In this section we introduce one...

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Unformatted text preview: Convex Sets In this section, we introduce one of the most important ideas in the theory of optimization, that of a convex set. We discuss other ideas which stem from the basic definition, and in particular, the notion of a convex function which will be important, for example, in describing appropriate constraint sets. The idea of a convex functions is usually first met in elementary calculus when the nec- essary conditions for a maximum or minimum of a differentiable function are discussed. There, one distinguishes between local maxima and minima by looking at the “concavity” of a function; functions have a local minimum at a point if they are “concave up” and a local maximum if they are concave down. The second derivative test is one which checks the concavity of the function. It is unfortunate that the terminology persists. As we shall see, the mathematical prop- erties of these functions can be fully described in terms of convex sets and in such a way that the analysis carries over easily from one to multiple dimensions. For these reasons, among others, we take, as our basic notion, the idea of a convex set in a vector space. Intuitively if we think of R 2 or R 3 , a convex set of vectors is a set that contains all the points of any line segment joining two points of the set (see the next figure). P Q Figure 1: A Convex Set P Q Figure 2: A Non-convex Set To be more precise, we introduce some definitions. Here, and in the following, we will always work in R n although the results are valid in any finite dimensional real vector space. 1 Definition 1.1 Let u, v ∈ R n . Then the set of all convex combinations of u and v is the set of points { w λ ∈ R n : w λ = (1 − λ ) u + λv, ≤ λ ≤ 1 } . (1.1) In, say, R 2 , this set is exactly the line segment joining the two points u and v . (See the examples below.) Next, is the notion of a convex set . Definition 1.2 Let K ⊂ R n . Then the set K is said to be convex provided that given two points u, v ∈ K the set (1.1) is a subset of K . We give some simple examples: Examples 1.3 (a) An interval of [ a, b ] ⊂ R is a convex set. To see this, let c, d ∈ [ a, b ] and assume, without loss of generality, that c < d . Let λ ∈ (0 , 1). Then, a ≤ c = (1 − λ ) c + λc < (1 − λ ) c + λd < (1 − λ ) d + λd = d ≤ b. (b) A disk with center (0 , 0) and radius c is a convex subset of R 2 . This may be easily checked by using the usual distance formula in R 2 namely bardbl x − y bardbl := radicalbig ( x 1 − y 1 ) 2 + ( x 2 − y 2 ) 2 and the triangle inequality bardbl u + v bardbl ≤ bardbl u bardbl + bardbl v bardbl . (Exercise!) (c) In R n the set H := { x ∈ R n : a 1 x 1 + . . . + a n x n = c } is a convex set. For any particular choice of constants a i it is a hyperplane in R n . Its defining equation is a generalization of the usual equation of a plane in R 3 , namely the equation ax + by + cz + d = 0....
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Convex Set - Convex Sets In this section we introduce one...

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