Perform one iteration of the optimal gradient steepest desce

Perform one iteration of the optimal gradient steepest descent method to locate the minimum of f(x,y)=-8x + x^2 +12y +4y^2 -2xy

using inital guesses x=0 and y=0

Solution

First iteration: x0 = 1, y0 = 1. f x|(1,1) = 2y + 2 2x|(1,1) = 6,

f y |(1,1) = 2x 4y|(1,1) = 6 f = 6~i 6~j g() = f(x0 + f x|(x0,y0)

· , y0 + f y |(x0,y0) · ) = f(1 + 6, 1 6) = 2 × (1 + 6) · (1 6) + 2(1 + 6) (1 + 6) 2 2(1 6) 2 = 180 2 + 72 7 g 0 () = 360 + 72 = 0, = 0.2. Second iteration: x1 = x0+ f x|(x0,y0) · = 1+6×0.2 = 0.2, y1 = y0+ f y |(x0,y0) · = 16×0.2 = 0.2 f x|(0.2,0.2) = 2y + 2 2x|(0.2,0.2) = 2 × (0.2) + 2 2 × 0.2 = 1.2, f y |(0.2,0.2) = 2x 4y|(0.2,0.2) = 2 × 0.2 4 × (0.2) = 1.2