Consider the unconstrained NLP

max x₁x₂ - 5(x₁-2)⁴ -3(x₂ - 5)⁴

once with the starting point of 1,3 and again with the starting point 4,0

(a) Use graphing software to produce a contour map of the objective function for x₁ ? [1,4], x₂ ?[2,8].

(b) Compute the move direction that would be pursued by gradient search Algorithm Step 0: Initialization. Choose any starting solution x ^{( 0 )}, pick stopping tolerance ? > 0, and set solution index t ? 0 .

Step 1: Gradient. Compute objective function gradient ? f ( x ^{( t )} ) at current point x ^.

Step 2: Stationary Point. If gradient norm ? ? ? f ( x ( t ) ) ? ? < ? ??f(x(t))?( t ) is sufficiently close to a stationary point.

Step 3: Direction. Choose gradient move direction ?x^(t+1)??f(x^(t))

( + for maximize and - for minimize).

Step 4: Line Search. Solve (at least approximately) corresponding one-dimensional line search max or min f ( x ^{( t )} + ? ? x ^{( t + 1 )} ) to compute ? _{t + 1}.

Step 5: New Point. Update

x ^{( t + 1 )} ? x ^{( t )} + ? _{t + 1} ? x ^{( t + 1 )}

Step 6: Advance. Increment t ? t + 1 , and return to Step 1.

Gradient norm ? ? f ( x ^{( t )} ) ? ? ? ? Squire root ? _j ( ? f /? x _i ) ² at x⁽⁰⁾=(1,3).

(c) State the line search problem implied by your direction.

(d)Solve your line search problem graphically and compute the next search point x⁽¹⁾.

(e) Do two additional iterations of Algorithm

Step 0: Initialization. Choose any starting solution x ^{( 0 )}, pick stopping tolerance ? > 0, and set solution index t ? 0

Step 1: Gradient. Compute objective function gradient ? f ( x ^{( t )} ) at current point x ^.

Step 2: Stationary Point. If gradient norm ? ? ? f ( x ( t ) ) ? ? < ? ??f(x(t))?( t ) is sufficiently close to a stationary point.

Step 3: Direction. Choose gradient move direction ?x^(t+1)??f(x^(t))

( + for maximize and - for minimize).

Step 4: Line Search. Solve (at least approximately) corresponding one-dimensional line search max or min f ( x ^{( t )} + ? ? x ^{( t + 1 )} ) to compute ? _{t + 1}.

Step 5: New Point. Update

x ^{( t + 1 )} ? x ^{( t )} + ? _{t + 1} ? x ^{( t + 1 )}

Step 6: Advance. Increment t ? t + 1 , and return to Step 1.

Gradient norm ? ? f ( x ^{( t )} ) ? ? ? ? Squire root ? _j ( ? f /? x _i ) ²

to compute x⁽²⁾ and x⁽³⁾

(f) Plot progress of the search on the contour map of part (a).