1. [6 marks] Determine whether the following function f on the convex set is convex, strictly...

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1. [6 marks] Determine whether the following function f on the convex set is convex, strictly convex, concave, strictly concave or neither: 1 1 f(x1, x2) = (x - 1) ex, N = {(x1, x2) R 2 : E 2. [17 marks] Consider the problem of minimizing the function f(x) = (4 x - x) + (x2 4) 0 on R2. Let x* = 2 i) Calculate the gradient f(x) and the Hessian (x) of f. ii) Show that x* is a stationary point off on R 2. iii) Find the other four stationary points off on R2 iv) Identify, as far as possible using Hessian information, the five stationary points of f of parts ii) and iii) as local minimizers local maximizers or saddle points, etc. v) Show that x* is a global minimizer of f. * , vi) Explain why x is not a strict global minimizer vii) Show that x* is a strict local minimizer off. of f. 3. [4 marks] Consider the quadratic function 9: Rn R defined by 1 1 q(x) = T (ATA)x (ATb) Tx + bTb, 2 2 where A is an n n matrix of rank n and b is a constant n 1 vector. Let x* = A-b. i) Write down the gradient v 9(x) and the Hessian 29(x) of 9(x). ii) Stating clearly any theorems that you use, show thatx* is the unique global min imizer of q(x). 4. [3 marks ] Let A be an n n matrix. Let A elements of A respectively, where i, j = {1,2, that A is indefinite and A be the ith and jth diagonal ii ny jj and i = j. If Au A < 0, then show jj 1. [6 marks] Determine whether the following function f on the convex set is convex, strictly convex, concave, strictly concave or neither: 1 1 f(x1, x2) = (x - 1) ex, N = {(x1, x2) R 2 : E 2. [17 marks] Consider the problem of minimizing the function f(x) = (4 x - x) + (x2 4) 0 on R2. Let x* = 2 i) Calculate the gradient f(x) and the Hessian (x) of f. ii) Show that x* is a stationary point off on R 2. iii) Find the other four stationary points off on R2 iv) Identify, as far as possible using Hessian information, the five stationary points of f of parts ii) and iii) as local minimizers local maximizers or saddle points, etc. v) Show that x* is a global minimizer of f. * , vi) Explain why x is not a strict global minimizer vii) Show that x* is a strict local minimizer off. of f. 3. [4 marks] Consider the quadratic function 9: Rn R defined by 1 1 q(x) = T (ATA)x (ATb) Tx + bTb, 2 2 where A is an n n matrix of rank n and b is a constant n 1 vector. Let x* = A-b. i) Write down the gradient v 9(x) and the Hessian 29(x) of 9(x). ii) Stating clearly any theorems that you use, show thatx* is the unique global min imizer of q(x). 4. [3 marks ] Let A be an n n matrix. Let A elements of A respectively, where i, j = {1,2, that A is indefinite and A be the ith and jth diagonal ii ny jj and i = j. If Au A < 0, then show jj