You are given the following function maxz x1 22 x1 x21 P

You are given the following function. maxz = -(x_1 - 2)^2 - x_1 - x_2^1 Plot the function with 3D function Plotter and past it to word file. Find the stationary point. Show that it is max. point. Use hessian. Use the method of steepest ascent to approximate the optimal solution to the above problem. Begin at the point (2.5, 1.5).

Solution

http://www.math.uri.edu/~bkaskosz/flashmo/graph3d/

or https://academo.org/demos/3d-surface-plotter/

(a) Use one of the online plotters (key in the function -(x-2)² -x-y² , say for x between 0 and 3 and y between -5 and 5)

(b)Let x and y be the variables x₁ and x₂.

Set F(x,y) =-(x-2)² -x-y²

Stationary point is obtained by setting the partial derivatives equal to 0. So we have

           -2(x-2) -1 =0 , x =1.5

                 -2y=0      , y=0

Now F_xx =-2, F_yy = -2 and F_xy =0. So this point is a maximum .

(c) Start from the point (2.5,1.5)

The gradient at thi²eepest ascent, set

G(t) = F(2.5-2t,1.5-3t) = -(0.5-t)² -(2.5-2t)-(1.5-3t)²

So G\'(t) = 4(0.5-2t)+2t+6(1.5-3t)

So G\'(t) =0 implies t =0.5

The new point (approximation is (1.5,0), at which the gradient vanishes.

So the method of steepest ascent terminates at (1.5,0) in one step.