Trust-region methods

Autor: Chun-Yu Chou (cc2398), Ting Guang Yeh (ty262), Yun-Chung Pan (yp392), Chen-Hua Wang (cw893), Fall 2021

Numerical example

Here we will use the trust-region method to solve a classic optimization problem, the Rosenbrock function. The Rosenbrock function is a non-convex function, introduced by Howard H. Rosenbrock in 1960, which is often used as a performance test problem for optimization algorithms. This problem is solved using MATLAB's fminunc as the solver, with 'trust-region' as the solving algorithm.

The function is defined by

$\min f(x,y)=100(y-x^{2})^{2}+(1-x)^{2}$

The starting point chosen is $x=0$ $y=0$.

Iteration Process

Iteration 1:

Summary of all iterations

Iteration	f(x)	x	y	Norm of step	First-order optimality
1	1	0.25	0	1	2
2	0.953125	0.263178	0.061095	0.25	12.5
3	0.549578	0.371152	0.124076	0.0625	1.63
4	0.414158	0.539493	0.262714	0.125	2.74
5	0.292376	0.608558	0.365573	0.218082	5.67
6	0.155502	0.765122	0.560477	0.123894	0.954
7	0.117347	0.804353	0.645444	0.25	7.16
8	0.0385147	0.804353	0.645444	0.093587	0.308
9	0.0385147	0.836966	0.69876	0.284677	0.308
10	0.0268871	0.90045	0.806439	0.0625	0.351
11	0.0118213	0.953562	0.90646	0.125	1.38
12	0.0029522	0.983251	0.9659	0.113247	0.983
13	0.000358233	0.99749	0.994783	0.066442	0.313
14	1.04121e-05	0.999902	0.999799	0.032202	0.0759
15	1.2959e-08	1	1	0.005565	0.00213
16	2.21873e-14	1	1	0.000224	3.59E-06