Trust-region methods: Difference between revisions
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=Problem formulation= | =Problem formulation= | ||
=Numerical example= | =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 | |||
<math>\min f(x,y)=100(y-x^2)^2+(1-x)^2</math> | |||
The starting point chosen is <math>x=0</math> <math>y=0 | |||
</math>. | |||
'''Iteration Process''' | |||
'''Iteration 1:''' | |||
{| class="wikitable" | |||
|+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 | |||
|} | |||
=Applications= | =Applications= | ||
=Conclusion= | =Conclusion= | ||
=References= | =References= |
Revision as of 01:12, 27 November 2021
Autor: Chun-Yu Chou (cc2398), Ting Guang Yeh (ty262), Yun-Chung Pan (yp392), Chen-Hua Wang (cw893), Fall 2021
Introduction
Problem formulation
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
The starting point chosen is .
Iteration Process
Iteration 1:
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 |