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Generalized Convexity and Optimization Lecture Notes in Economics and Mathematical Systems 616 Generalized Convexity and Optimization Theory and Applications Bearbeitet von Alberto Cambini, Laura Martein 1. Auflage 2008. Taschenbuch. xiv, 248 S. Paperback ISBN 978 3 540 70875 9 Format (B x L): 15,5 x 23,5 cm Gewicht: 405 g Wirtschaft > Betriebswirtschaft: Theorie & Allgemeines > Wirtschaftsmathematik und - statistik Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. 2 Non-Differentiable Generalized Convex Functions 2.1 Introduction In several economic models convexity appears to be a restrictive condition. For instance, classical assumptions in Economics include the convexity of the production set in producer theory and the convexity of the upper level sets of the utility function in consumer theory. On the other hand, in Optimization it is important to know when a local minimum (maximum) is also global. Such a useful property is not exclusive to convexity (concavity). All this has led to the introduction of new classes of functions which have con- vex lower/upper level sets and which verify the local-global property, starting with the pioneer work of Arrow–Enthoven [7]. In this chapter we shall introduce the class of quasiconvex, strictly quasicon- vex, and semistrictly quasiconvex functions and the inclusion relationships between them are studied. For functions in one variable, a complete charac- terization of the new classes is established. Examples of the most important generalized convex functions in Economics are given. We shall focus our attention on generalized convexity since a function f is gen- eralized concave if and only if −f is generalized convex, so that every result for a generalized convex function can be easily re-stated in terms of general- ized quasiconcave functions. For the sake of completeness in Appendix B we shall give a summary of the main properties of generalized concave functions. 2.2 Quasiconvexity and Strict Quasiconvexity One way to generalize the definition of a convex function is to relax the convexity condition, and require, from a geometrical point of view, that the restriction of the function along a line joining any two points in the domain lies under at least one of the endpoints. A function that verifies such a con- dition is called quasiconvex. Figure 2.1 shows some examples of quasiconvex functions. 24 2 Non-Differentiable Generalized Convex Functions x x 2 2 x2 0 x1 0 123x 1 0 x1 Fig. 2.1. Quasiconvex functions From an analytical point of view, we have the following definition. Definition 2.2.1. Let f be defined on a convex set S ⊆n. The function f is said to be quasiconvex on S if f(λx1 +(1− λ)x2) ≤ max{f(x1),f(x2)} (2.1) for every x1,x2 ∈ S and for every λ ∈ [0, 1] or, equivalently, f(x1) ≥ f(x2) implies that f(x1) ≥ f(x1 + λ(x2 − x1)) (2.2) for every x1,x2 ∈ S and for every λ ∈ [0, 1]. If the inequality in (2.1) is strict, the function is called strictly quasiconvex. Formally: Definition 2.2.2. Afunctionf defined on a convex set S ⊆n is said to be strictly quasiconvex if f(λx1 +(1− λ)x2) < max{f(x1),f(x2)} (2.3) for every x1,x2 ∈ S, x1 = x2, and for every λ ∈ (0, 1) or, equivalently, f(x1) ≥ f(x2) implies that f(x1) >f(x1 + λ(x2 − x1)) (2.4) for every x1,x2 ∈ S, x1 = x2, and for every λ ∈ (0, 1). It follows immediately, from the given definitions, that a strictly quasicon- vex function is also quasiconvex; the converse is not true as is shown in the following example. Example 2.2.1. | | x x =0 1. The function f(x)= x is quasiconvex but not strictly quasi- 0 x =0 convex; 2. Every monotone function of one variable is quasiconvex and every increas- ing or decreasing function of one variable is strictly quasiconvex; for instance, the concave function f(x)=logx, x > 0 is strictly quasiconvex. 2.2 Quasiconvexity and Strict Quasiconvexity 25 As we have just pointed out, quasiconvexity may be viewed as a relaxation of convexity. Unfortunately, in the new larger class of functions some proper- ties of convex functions are lost. For instance, as we can deduce from (1) in Example 2.2.1, a quasiconvex function may have interior discontinuity points, a local minimum which is not global and an interior global maximum. The relationships between convexity, strict convexity, quasiconvexity and strict quasiconvexity are stated in the following theorem. Theorem 2.2.1. Let S ⊆n be a convex set. (i) If f is convex on S,thenf is quasiconvex on S. (ii) If f is strictly convex on S,thenf is strictly quasiconvex on S. (iii) If f is strictly quasiconvex on S,thenf is quasiconvex on S. Proof. (i) We have: f(λx1 +(1− λ)x2) ≤ λf(x1)+(1− λ)f(x2) ≤ λ max{f(x1),f(x2)} +(1− λ) max{f(x1),f(x2)} =max{f(x1),f(x2)}. Similarly we can prove (ii), while (iii) follows directly by definition. Example 2.2.1 shows that the class of quasiconvex functions contains properly the class of convex functions and the class of strictly quasiconvex functions. Furthermore, there is not any inclusion relationship between the class of con- vex functions and the class of strictly quasiconvex ones. In fact, a convex function may have constant restrictions and a strictly increasing function of one variable is strictly quasiconvex but not necessarily convex. The relationships between convexity, strict convexity, quasiconvexity and strict quasiconvexity are illustrated in Fig. 2.2 where the arrow (→)reads “implies”. strictly strictly convex quasiconvex convex quasiconvex Fig. 2.2. Relationships between various types of convexity An important case where quasiconvexity reduces to convexity is related to homogeneity. More exactly, the homogeneity assumption combined with quasiconvexity produces convexity as is shown in the following theorem. Theorem 2.2.2. Let f be a homogeneous function of degree α =1defined on a convex set S ⊆n. If f(x) > 0 for all x ∈ S,thenf is quasiconvex if and only if it is convex. 26 2 Non-Differentiable Generalized Convex Functions Proof. From (i) of Theorem 2.2.1, convexity implies quasiconvexity without any other assumption. Now we will prove, firstly, that homogeneity combined with quasiconvexity implies subadditivity, i.e., for every x1,x2 ∈ S we have f(x1 + x2) ≤ f(x1)+f(x2). Let y1 = f(x1) > 0, y2 = f(x2) > 0. Since f is homogeneous of degree one, i.e., f(tx)=tf(x),t > 0, we have f( x1 )=f( x2 ) = 1, so that the quasicon- y1 y2 vexity of f implies f((1 − t) x1 + t x2 ) ≤ 1 ∀t ∈ (0, 1). By choosing t = y2 , y1 y2 y1+y2 we have 1 − t = y1 ,andf( x1 + x2 )= 1 f(x + x ) ≤ 1. Conse- y1+y2 y1+y2 y1+y2 y1+y2 1 2 quently, f(x1 + x2) ≤ (y1 + y2)=f(x1)+f(x2). It remains to be proven that subadditivity and homogeneity imply convexity. We have f(λx1 +(1−λ)x2) ≤ f(λx1)+f((1−λ)x2)=λf(x1)+(1−λ)f(x2). While a convex function may be characterized by the convexity of its epi- graph, a quasiconvex function may be characterized by the convexity of its lower level sets, as is shown in the following theorem. Theorem 2.2.3. Afunctionf defined on a convex set S ⊆n is quasiconvex on S if and only if the lower level set L≤α={z ∈ S : f(z) ≤ α} is convex for every α ∈. Proof. Assume that f is quasiconvex and let x, y ∈ L≤α.Sincef(x) ≤ α, f(y) ≤ α,wehavef(λx +(1− λ)y) ≤ max{f(x),f(y)}≤α,sothat λx +(1− λ)y ∈ L≤α. In order to prove the converse statement, assume without loss of general- ity, that max{f(x),f(y)} = f(x) and consider the lower level set L≤α with α = f(x), that is L≤f(x) = {z ∈ S : f(z) ≤ f(x)}; obviously y ∈ L≤f(x). Since L≤f(x) is convex, x + λ(y − x) ∈ L≤f(x) for every λ ∈ [0, 1], i.e., f(x + λ(y − x)) ≤ f(x)=max{f(x),f(y)}. By means of a simple application of the previous Theorem, we obtain the following result. Theorem 2.2.4. Let f be a quasiconvex function defined on a convex set S ⊆n and let S∗ be the set of all global minimum points of f.Then,S∗ is convex. Proof. If S∗ = ∅ the thesis follows by convention. Taking into account that S∗ = {x ∈ S : f(x)=m} = {x ∈ S : f(x) ≤ m},wherem is the minimun value of f on S, the convexity of S∗ follows from the convexity of the lower level sets of a quasiconvex function. As happens for a convex function and for a strictly convex function, (2.1) and (2.3) may be extended to every convex combination of a finite number of points. Theorem 2.2.5. f is quasiconvex on a convex set S ⊆n if and only if, for xi ∈ S, i =1, .., p, we have 2.2 Quasiconvexity and Strict Quasiconvexity 27 p p i i f λix ≤ max f(x ), λi =1,λi ≥ 0,i=1, .., p. (2.5) i∈{1,..,p} i=1 i=1 Furthermore, f is strictly quasiconvex on S if and only if the inequality in (2.5) is strict.
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