4. Convex Sets and (Quasi-)Concave Functions

Transcription

1 4. Convex Sets and (Quasi-)Concave Functions Daisuke Oyama Mathematics II April 17, 2017

2 Convex Sets Definition 4.1 A R N is convex if (1 α)x + αx A whenever x, x A and α [0, 1]. A R N is strictly convex if (1 α)x + αx Int A whenever x, x A, x x, and α (0, 1). 1 / 32

3 Convex Combinations For α 1,..., α M 0, M m=1 α m = 1, α 1 x α M x M is called a convex combination of x 1,..., x M. Proposition 4.1 If A R N is convex, then any convex combination of elements in A is contained in A. Proof By induction. 2 / 32

4 Convex Hull Proposition 4.2 For any index set Λ, if C λ R N is convex for all λ Λ, then λ Λ C λ is convex. (The intersection of any family of convex sets is convex.) Definition 4.2 For A R N, the convex hull of A, denoted CH A, is the intersection of all convex sets that contain A (or, the smallest convex set that contains A). 3 / 32

5 Proposition 4.3 For A R N, CH A equals the set of all convex combinations of elements in A, i.e., CH A = { x R N x = M α m x m m=1 for some M N, x 1,..., x M A, and M } α 1,..., α M 0 with α m = 1. m=1 4 / 32

6 Algebra of Convex Sets Proposition 4.4 If A, B R N are convex, then A + B = {x R N x = a + b for some a A and b B} is convex. If A R N is convex, then for t R, ta = {x R N x = ta for some a A} is convex. 5 / 32

7 Convex Cones Definition 4.3 A R N is a cone if x A αx A for any α 0. A R N is a convex cone if x, y A αx + βy A for any α, β 0. (Some textbooks define with for any α > 0 and for any α, β > 0.) 6 / 32

8 Carathéodory s Theorem For A = {x 1,..., x M } R N, write Cone A = { M m=1 α mx m R N α 1,..., α M 0}, where M m=1 α mx m, α m 0, is called a conic combination of x 1,..., x M R N. Proposition 4.5 (Carathéodory s Theorem) 1. For A = {x 1,..., x M } R N, A {0}, each x Cone A is written as a conic combination of linearly independent elements of A. 2. For A R N, each x CH A is written as a convex combination of at most N + 1 elements in A. 7 / 32

9 Proof 1. Let x = α 1 x α M x M, (1) where we assume without loss of generality that α 1,..., α M > 0. If x 1,..., x M are linearly independent, we are done. Suppose that x 1,..., x M are linearly dependent, so that c 1 x c M x M = 0 (2) for some (c 1,..., c M ) (0,..., 0). Assume that c m > 0 for some m (if c m 0 for all m, then multiply both sides by 1). { Let µ = min αm } cm c m > 0 > 0. 8 / 32

10 Proof By (1) and (2) we have where x = (α 1 µc 1 )x (α M µc M )x M, α m µc m 0 for all m, and αm µc m = 0 for some m. Thus x has been written as a conic combination of M 1 (or fewer) elements of {x 1,..., x M }. If these M 1 (or fewer) vectors are linearly independent, we are done. If not, repeat the same procedure. With finitely many steps, we have a conic representation with linearly independent vectors. 9 / 32

11 Proof 2. For A R N, let x CH A. Then we have x = α 1 x α M x M for some x 1,..., x M A and some α 1,..., α M 0 with α α M = 1. Consider B = {(x 1, 1),..., (x M, 1)} R N+1. Then (x, 1) Cone B. By part 1, there are linearly independent elements {(x m 1, 1),..., (x m K, 1)} from B such that (x, 1) = β 1 (x m 1, 1) + + β K (x m K, 1), where β k 0 and K N + 1. From the 1st through Nth coordinates we have x = β 1 x m β K x m K, while from the (N + 1)st coordinate we have β β K = / 32

12 Topological Properties of Convex Sets Lemma 4.6 Let A R N be a convex set. If x Int A and y Cl A, then (1 α)x + αy Int A for all α [0, 1). Proposition 4.7 Let A R N be a convex set. Then Int A and Cl A are convex. Proof Int A: By Lemma 4.6. Cl A = ε>0 (A + N ε(0)). 11 / 32

13 Topological Properties of Convex Sets Fact 1 Let A R N be a convex set. If Int(Cl A), then Int A. Proposition 4.8 Let A R N be a convex set. Then Int(Cl A) = Int A. 12 / 32

14 Topological Properties of Convex Sets Proposition 4.9 If A R N is bounded, then Cl(CH A) = CH(Cl A). In particular, if A is compact, then CH A is compact. 13 / 32

15 Proof Since CH A A, we have Cl(CH A) Cl A. Since Cl(CH A) is convex (Proposition 4.8), we have Cl(CH A) CH(Cl A). Since A Cl A, we have CH A CH(Cl A). We want to show that CH(Cl A) is closed if A is bounded. Let {x m } CH(Cl A), and assume x m x. By Carathéodory s Theorem (Proposition 4.5(2)), each x m is written as where x m = α m 1 x m,1 + + α m N+1x m,n+1, (α m 1,..., α m N+1 ) = {α RN+1 α n 0, n α n = 1}, x m,1,..., x m,n+1 Cl A. 14 / 32

16 Proof Since and Cl A are compact, there exists a sequence {m(k)} such that the limits ᾱ n = lim k αn m(k) and x n = lim k x m(k),n exists where (ᾱ 1,..., ᾱ N+1 ) and x 1,..., x N+1 Cl A. Hence, x = ᾱ 1 x ᾱ N+1 x N+1, so that x CH(Cl A). 15 / 32

17 Concave Functions Definition 4.4 Let X R N be a non-empty convex set. A function f : X R is concave if f((1 α)x + αx ) (1 α)f(x) + αf(x ) for all x, x X and all α [0, 1]. f : X R is strictly concave if f((1 α)x + αx ) > (1 α)f(x) + αf(x ) for all x, x X with x x and all α (0, 1). f : X R is convex (strictly convex, resp.) if f is concave (strictly concave, resp.). 16 / 32

18 Hypograph and Epigraph Let X R N be a non-empty set. The hypograph of a function f : X R is the set hyp f = {(x, y) R N R x X, y f(x)}. The epigraph of a function f : X R is the set epi f = {(x, y) R N R x X, y f(x)}. Proposition 4.10 Let X R N be a nonempty convex set. f : X R is a concave (convex, resp.) function if and only if hyp f (epi f, resp.) is a convex set. 17 / 32

19 Jensen s Inequality Proposition 4.11 Let X R N be a nonempty convex set. If f : X R is concave, then f(α 1 x α M x M ) α 1 f(x 1 ) + + α M f(x M ) for any x 1,..., x M X and α 1,..., α M 0 with M m=1 α m = 1. Proposition 4.12 Let I R be a nonempty closed interval. If f : I R is concave, then ( ) f x df (x) f(x) df (x) for any distribution function F on I. 18 / 32

20 Properties of Concave Functions Let X R N be a non-empty convex set. Lemma 4.13 f : X R is (strictly) concave if and only if for any x X and any z R N with x + z X, for t (0, 1], f(x + tz) f(x) t is nonincreasing (strictly decreasing) in t. Proof If t < t with t = αt, α (0, 1), then we have f(x + t z) (1 α)f(x) + αf(x + tz) f(x + t z) f(x) αt f(x + tz) f(x). t 19 / 32

21 Continuity of Concave Functions Let X R N be a non-empty convex set. Lemma 4.14 Let f : X R be a concave function. If x Int X, then there exist ε > 0 and M such that f(x) M for all x N ε ( x). Proposition 4.15 A concave function f : X R is continuous on Int X. 20 / 32

22 Proof of Lemma 4.14 Let x Int X. Let δ > 0 be such that N δ ( x) X, and let ε = δ/ N. Let S = {x R N x x ε} N δ ( x). Let v 1,..., v m be the m = 2 N vertices of S (so that S = CH{v 1,..., v m }). Let L = min{f(v 1 ),..., f(v m )}. Then f(x) L for all x S by the concavity of f. Take any x N ε ( x), and let y N ε ( x) be such that x = 1 2 x y. Since f( x) 1 2 f(x) + 1 2f(y), we have f(x) 2f( x) f(y) 2f( x) L. Finally, let M = max{ L, 2f( x) L }. 21 / 32

23 Proof of Proposition 4.15 Let x Int X. By Lemma 4.14, we can take r > 0 and M such that f(x) M for all x N 2r ( x). Take any x, y N r ( x). We want to show that f(y) f(x) 2M r y x. Let z = x + y x +r y x (y x). Then z N 2r ( x). 22 / 32

24 By Lemma 4.13, we have y x f(y) f(x) (f(z) f(x)) y x + r y x f(z) f(x) y x + r y x f(z) f(x) r y x ( f(z) + f(x) ) r y x 2M. r By a symmetric argument, we have x y f(x) f(y) 2M. r 23 / 32

25 Extended Real Valued Functions Let X R N be a nonempty convex set. Definition 4.5 A function f : X (, ] is defined to be convex if f((1 α)x + αx ) (1 α)f(x) + αf(x ) for all x, x X and all α [0, 1], where α = if α > 0, 0 = 0, + y = y + = for y (, ], and y for y (, ]. (Concavity of a function f : X [, ) is defined analogously.) 24 / 32

26 Extended Real Valued Functions Let X R N be a nonempty convex set. Proposition 4.16 A function f : X (, ] is convex if and only if epi f is a convex set. Any convex function f : X R can be extended to R N keeping convexity, by assigning to x / X. 25 / 32

27 Convex Optimal Value Functions Let X R N be a nonempty set, and let P R M be a nonempty convex set. Proposition 4.17 Consider a function f : X P R. If for all x X, f(x, p) is convex in p, then the function v : P (, ] defined by v(p) = sup f(x, p) x X is convex. Proof Show that epi v is a convex set. ( Homework) 26 / 32

28 Support Functions For a nonempty set A R N, the function ϕ A : R N (, ] defined by ϕ A (p) = sup p x x A is called the support function of A. The profit function is the support function of the production set (but only defined for nonnegative/positive price vectors). The cost function is the concave support function of the input requirement set (Section 9.9), which is defined with inf in place of sup. The expenditure function is the concave support function of the upper utility level set (Chapter 10). 27 / 32

29 Support Functions Proposition 4.18 The support function ϕ A : R N (, ] is 1. convex and 2. homogeneous of degree one, i.e., For all p R N, ϕ A (tp) = tϕ A (p) for all t > 0. Proof 1. By Proposition For all x A, (tp) x t supx A p x, so sup x A (tp) x t sup x A p x. For all x A, supx A(tp) x t(p x), so (1/t) sup x A(tp) x sup x A p x. 28 / 32

30 Quasi-Concave Functions Definition 4.6 Let X R N be a non-empty convex set. f : X R is quasi-concave if f((1 α)x + αx ) f(x ) for all x, x A such that f(x) f(x ) and all α [0, 1]. f : X R is strictly quasi-concave if f((1 α)x + αx ) > f(x ) for all x, x A with x x such that f(x) f(x ) and all α (0, 1). f : X R is semi-strictly quasi-concave if f((1 α)x + αx ) > f(x ) for all x, x A such that f(x) > f(x ) and all α (0, 1). f is quasi-/strictly quasi-/semi-strictly quasi-convex if f is quasi-/strictly quasi-/semi strictly quasi-concave. 29 / 32

31 Equivalent Definition Proposition 4.19 f : X R is quasi-concave if and only if {x X f(x) t} is convex for all t R. 30 / 32

32 Properties of Quasi-Concave Functions Let X R N be a non-empty convex set. Proposition 4.20 If f : X R is quasi-concave (strictly quasi-concave) and h: R R is nondecreasing (strictly increasing), then h f is quasi-concave (strictly quasi-concave). 31 / 32

33 Properties of Quasi-Concave Functions Let X R N be a non-empty convex set. For f : X R, write X = {x X f(x) = sup x X f(x )}. Proposition If f is quasi-concave, then X is a convex set. 2. If f is strictly quasi-concave, then X is either empty or a singleton set. 32 / 32