NUMERICAL OPTIMISATION
TUTORIAL 2
MARTA BETCKE
KIKO RUL·LAN
EXERCISE 1
(a) Code backtracking line search, steepest descent and Newton’s algorithms. See Cody Courseworks
for more guidance.
Submit your implementation via Cody Coursework. [30pt]
(b) Apply steepest descent and Newton’s algorithms (with backtracking line search) to minimise the
Rosenbrock function
f(x) = 100(y − x2)2 + (1 − x)2.
Set the initial point x0 = (1.2, 1.2)
T and the initial step length α0 = 1. Plot the step sizes used
by each method over the iterations as well as the trajectories traced by the iterates in R2. Try
explaining the trajectories.
Submit solution via TurnitIn. [40pt]
(c) Redo the calculations in b) with the more difficult starting point x0 = (−1.2, 1)T and explain the
trajectories.
Submit solution via TurnitIn. [10pt]
(d) Repeat the calculations in b) and c) using the line search in Algorithm 3.5 from Nocedal, Wright.
This line search produces step lengths which satisfy the strong Wolfe conditions. Use the imple-
mentation provided in Moodle: lineSearch.m, zoomInt.m. Compare the new step lengths with
those obtained with backtracking.
Submit solution via TurnitIn. [20pt]
Remark. The submission to TurnitIn should not be longer than 4 pages. Avoid submitting more
code than needed (if any) and focus on explaining your results.
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