(a) Let A be a mx n matrix, b a m-dimensional vector, i.e., A Rmxn, be...

  

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(a) Let A be a mx n matrix, b a m-dimensional vector, i.e., A Rmxn, be Rm, and f: Rm → R be a convex function. Show that g(x) = f(Ar+b) is a convex function. Note that g: R" → R. (b) (5%) Show that the function h: R→ R, given by h(x) = ln(1 + e*) is convex. (c) (10%) Consider the function f : R² → R, f(x1,2)= ln(e1 + e²2) Using the result from (a) and (b), or otherwise, show that f(x₁, x2) is convex. (d) (5%) Write down an optimality condition that characterizes an optimal solution (x,x) to the following problem min ⓇIn(e²¹ + €²²) + 2/2a²³² + 21,72ER You are not required to provide an exact answer to (1,2). ¹This fact comes from a direct application of the Cauchy-Schwarz inequality for random variables, i.e., E[XY] ≤ √E[X²]E[Y²]. (a) Let A be a mx n matrix, b a m-dimensional vector, i.e., A Rmxn, be Rm, and f: Rm → R be a convex function. Show that g(x) = f(Ar+b) is a convex function. Note that g: R" → R. (b) (5%) Show that the function h: R→ R, given by h(x) = ln(1 + e*) is convex. (c) (10%) Consider the function f : R² → R, f(x1,2)= ln(e1 + e²2) Using the result from (a) and (b), or otherwise, show that f(x₁, x2) is convex. (d) (5%) Write down an optimality condition that characterizes an optimal solution (x,x) to the following problem min ⓇIn(e²¹ + €²²) + 2/2a²³² + 21,72ER You are not required to provide an exact answer to (1,2). ¹This fact comes from a direct application of the Cauchy-Schwarz inequality for random variables, i.e., E[XY] ≤ √E[X²]E[Y²].