The Steepest Descent Algorithm for Unconstrained Optimization and ...

• x 0 =(0, 0) T .
• x 0 =(−0.4, 0) T .
• x 0 =(10, 0) T .
• x 0 =(11, 0) T .
As it turns out, the optimal solution to this problem is x =(5, 6) T ,
∗
with f(x )= −22. What linear convergence constants do you observe
for each of the above starting points
4. (Steepest Descent) Suppose we seek to minimize
1 T
f(x) = x T Hx + c x,
2
where ⎛ ⎞ ⎛ ⎞
10 −18 2 12
⎜ ⎟ ⎜ ⎟
H = ⎝ −18 40 −1 ⎠ and c = ⎝ −47 ⎠ .
2 −1 3 −8
Implement the steepest descent algorithm on this problem, using the
following starting points.
• x 0 =(0, 0, 0) T .
• x 0 =(15.09, 7.66, −6.56) T .
• x 0 =(11.77, 6.42, −4.28) T .
• x 0 =(4.46, 2.25, 1.85) T .
As it turns out, the optimal solution to this problem is x =(4, 3, 1) T ,
∗
with f(x )= −50.5. What linear convergence constants do you observe
for each of the above starting points
5. (Steepest Descent) Suppose that we seek to minimize the following
function:
f(x 1 ,x 2 )= −9x 1 −10x 2 +θ(− ln(100−x 1 −x 2 )−ln(x 1 )−ln(x 2 )−ln(50−x 1 +x 2 )),
where θ is a given parameter. Note that the domain of this function
is X = {(x 1 ,x 2 ) | x 1 > 0,x 2 > 0,x 1 + x 2 < 100,x 1 − x 2 < 50}.
Implement the steepest descent algorithm for this problem, using the
∗
∗
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