Question

The steepest descent method for minimize f(x) is the gradient descent method using exact line search, that is, the step size of the kth iteration is chosen as ak = argmin f(xk – aVf(xk)). a>0 (a) (3 points) Consider the objective function f(x): = *Ax – cx+d, where A e Rnxn, CER”, d E R are given. Assume that A is symmetric positive definite and, at xk, Vf(xk) + 0. Give a formula of ak in terms xk, A, c, d. (b) (5 points) Consider R2. Starting from xº = (0,0), perform two iterations of the steepest descent method for minimize f(x1, x2) $(x{ + až) + 2x2 +1. 1 =

Answer 1

Answer:

@ Given, f(x) = 1/2" Ax- c'x+d, f: IR" IR that is differentiable at a function to , the dixeetion of steepest descent is the neeter sel take the the finding of (xo) TO o(f) = flottu) u ih unit neuter = 2 of of 10 X, st + at ann af af un ut Afro+Lulu Blo)= Atxo) u 1187 (20) 11 coro floo) الامت ۱۷|

O'(o)= -119 f(x0)11 define d. (t) = f(xo-* vf(xo)) x1=xo - to f(xo) and then continue the process by Storching from a in the direction of of (x1) to obtain als by the minimising de (+) = f(x, - tof(x1) and 90 on- an initial This Hethod of sentence descent given the Method computer d a sequence of interatu {xx} where as, guess no Xx+1 = Xx-te of (x), k=0, 2... where tieso Hinimies the function. de (t) = flaunt of (xi)) Given, Minimise f(x1,Kg) = ½ Cxi +xz) + 2x, to - The starting from a = (0,0) Minimise f(x, x2) = 12 (x12 +28) + 2R2+) I f(x, X₂) = bu x2+2) If (xo) = (0,2) det us Mininise the function

Ø (t) = f(10.o)t(0,2)) flti-at) on computing d' (t) = - Delt, -2 t) Cor) - (-t , -at+2).(0,2) (0, -ut +4) cet - 4 uttico t=1 &"(t) = USO & Minimas x = x₂ - 1/2 of (xol = (0,0) - 1/2 Coa) x1=X0-1 f(xo) = (oo) - 1(0,2) = (0, 2) det's continuing this process then that we have VF(x) = f(0, 2) =(0,0) 8 (t) = f(0, 2) - 110,0) f(0, -2) = 1/2 (oth) -+ (-U) +1 رمه ملم 2-uri constant so we stops our inter alion of function t then proceed ar above

I hope you understood my answer. Please give Up vote(LIKE) my answer. Thank you......