TEDU - MATH 502 - NUMERICAL OPTIMIZATION
Written Homework 1 (Deadline : 28.04.2023, 12:00pm )
1) Let
1
minimize f (x) = xT Qx − cT x
2


 
9 3 1
16



with Q = 3 7 2 , c = 0  and an initial guess x0 = (0, 0, 0)T be given.
1 2 5
18
a) Apply the Newton’s method for minimization (with step length α = 1 at each iterations).
b) Apply the steepest descent method.
c) Apply symmetric rank-one update formula with B0 = I.
d) Apply BFGS quasi-Newton method with B0 = I.
In parts (b),(c),(d), use an exact line search to determine the step length αk .
Perform at least 2 iterations by hand.
Implementation of each parts (a),(b),(c),(d) in a script and printing the results are required.
2) Consider the problem
minimize
subject to
f (x) = x21 + x21 x23 + 2x1 x2 + x42 + 8x2
3x1 − x2 + x3 = 3
Show that the point (−1, −1, −1) is a stationary point. Determine whether this point (−1, −1, −1)
is a local minimizer or not.