For an optimization problem min it] . [I] J. the optimal solution is D and the optimal value is I]. With a constant step size n r: I], the gradient method is 11\"] = 11"" 1:1?frkl (El and the proximal minimisation1 or the proximal point algorithm [PPAl is 15"\" = Proxufrkl. {3} Each method generates a sequence {.rk} based on a sequence update equation We are interested in whether the sequence converges to a limit point that is the optima] solution of Eq. If I]. Note that a sequence generated by .r"+1 = cur" converges to [l if |c| c: 1. We consider fir] = rrdr, where .4 >- H and its gradient 'xf'r] = .41. For simple case, we suppose .4 = diag{t1].og.~- .on] with [t c at 5 c2 5 E r1,21 where diag[c1.o2. - -- .orl] is a diagonal matrix that has 11.- in the 1-th diagonal element. Then. we have f[;1*} = %Ef=lr1..r? and all j ?2f|{.r] = .4 j owl. You need to app|_v the gradient method and the proximal point algorithm to solve Eq. [ I}. l. Derive a sequence update equation using the gradient method in [2]. What is the condition of n: such that the sequence generated h_v the gradient method converges to the limit. 2. Derive a sequence update equation using the proximal point algorithm in {3}. What is the condition of n: such that the sequence generated h_v the proximal point algorithm converges to the limit. 3. Discuss the convergence of the algorithms in terms of 11. Which method converges in a larger range of 121'? For the same a 3' it. which method converges faster