question archive 1) [6 marks ] Determine whether the following function f on the convex set Q is convex, strictly convex, concave, strictly concave or neither: 1 1 f(a:1,$2)=_(m1_1)2e"§, Q={(m1,mz)€l&2 :?sm ESE}- [17 marks ] Consider the problem of minimizing the function f(x) = (4— $3 — $32 + ($3 — 4)2 0 2 i) Calculate the gradient Vf(x) and the Hessian V2f(x) of f
1) [6 marks ] Determine whether the following function f on the convex set Q is convex, strictly convex, concave, strictly concave or neither: 1 1 f(a:1,$2)=_(m1_1)2e"§, Q={(m1,mz)€l&2 :?sm ESE}- [17 marks ] Consider the problem of minimizing the function f(x) = (4— $3 — $32 + ($3 — 4)2 0 2 i) Calculate the gradient Vf(x) and the Hessian V2f(x) of f
Subject:MathPrice: Bought3
1) [6 marks ] Determine whether the following function f on the convex set Q is convex, strictly convex, concave, strictly concave or neither: 1 1 f(a:1,$2)=_(m1_1)2e"§, Q={(m1,mz)€l&2 :?sm ESE}- [17 marks ] Consider the problem of minimizing the function f(x) = (4— $3 — $32 + ($3 — 4)2 0 2 i) Calculate the gradient Vf(x) and the Hessian V2f(x) of f. on RZ. Let x* = Show that x* is a stationary point off on R2. I: 11 >—- 1 Identify, as far as possible using Hessian information, the five stationary points of <,' ) ) Find the other four stationary points of f on R2. ) f of parts ii) and iii) as local minimizers , local maximizers or saddle points , etc. v) Show that x* is a global minimizer off. vi) Explain why x* is not a strict global minimizer off. vii) Show that x* is a strict local minimizer off. [4 marks ] Consider the quadratic function q : R" _) R defined by 1 1 q(x) = §xT(ATA)x _ (ATb)Tx + 5bTb, where A is an n n matrix of rank n and b is a constantn X 1 vector. Let x* = A—lb. X i) Write down the gradient Vq(x) and the Hessian V2q(x) of q(x). ii) Stating clearly any theorems that you use, show thatx* is the unique global min - / imizer of q(x). [3 marks ] Let A be an n x n matrix . Let As and A" be the W" and j"ill diagonal elements of A respectively, where 133' 6 {1,2, . . . , n} and 2' = j. IfA?Ajj < 0, then show that A is indefinite
