Suppose that f(x, y) is convex with respect to a. Prove that the function Fa(x,C) is...

  

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Suppose that f(x, y) is convex with respect to a. Prove that the function Fa(x,C) is convex with respect to x. Problem 2 (40 points) Prove Problem 3 (20 points) Prove Sa(a) arg min F(x,C) a(x) = min Fa(x,() Hint 1 Take into account that the following statement is valid, ac - | E{{f(x, y) (\+} = | | \f(x, z) C]v(z) dz a = 0 (x, z) > < (f(x, z) () v(z) dz ac = (x2) 2017 (f(x a =(x2) (f(x, z) - () v(z) dz (-1) v(z) dz Hint 2 First, prove that Fa(, ) is convex with respect to by using Definition 2. Second, equate (a, c) to zero and find VaR. Finally, place VaR expression to function Fa(x. C) and find CVaR. Loss is defined as a function f(x, y) of a decision vector x = X CR" and a random vector y = YC Rm. Definition 1 (CDF) We suppose that random vector y has a multidimensional density function v(-) and cumulative distribution function (CDF) for the random loss f(x, y) is equal to V(rc)=P{y|f(x,y) } = {(x) < Suppose that f(x, y) is convex with respect to a. Prove that the function Fa(x,C) is convex with respect to x. Problem 2 (40 points) Prove Problem 3 (20 points) Prove Sa(a) arg min F(x,C) a(x) = min Fa(x,() Hint 1 Take into account that the following statement is valid, ac - | E{{f(x, y) (\+} = | | \f(x, z) C]v(z) dz a = 0 (x, z) > < (f(x, z) () v(z) dz ac = (x2) 2017 (f(x a =(x2) (f(x, z) - () v(z) dz (-1) v(z) dz Hint 2 First, prove that Fa(, ) is convex with respect to by using Definition 2. Second, equate (a, c) to zero and find VaR. Finally, place VaR expression to function Fa(x. C) and find CVaR. Loss is defined as a function f(x, y) of a decision vector x = X CR" and a random vector y = YC Rm. Definition 1 (CDF) We suppose that random vector y has a multidimensional density function v(-) and cumulative distribution function (CDF) for the random loss f(x, y) is equal to V(rc)=P{y|f(x,y) } = {(x) <