In this example we want to use AlgoPy to help compute the minimum of the non-convex bivariate Rosenbrock function

\[f(x, y) = (1 - x)^2 + 100 (y - x^2)^2\]

The idea is that by using AlgoPy to provide the gradient and hessian of the objective function, the nonlinear optimization procedures in scipy.optimize will more easily find the \(x\) and \(y\) values that minimize \(f(x, y)\). Here is the python code:

```
"""
Minimize the Rosenbrock banana function.
http://en.wikipedia.org/wiki/Rosenbrock_function
"""
import numpy
import minhelper
def rosenbrock(X):
"""
This R^2 -> R^1 function should be compatible with algopy.
http://en.wikipedia.org/wiki/Rosenbrock_function
A generalized implementation is available
as the scipy.optimize.rosen function
"""
x = X[0]
y = X[1]
a = 1. - x
b = y - x*x
return a*a + b*b*100.
def main():
target = [1, 1]
easy_init = [2, 2]
hard_init = [-1.2, 1]
minhelper.show_minimization_results(
rosenbrock, target, easy_init, hard_init)
if __name__ == '__main__':
main()
```

Here is its output:

```
properties of the function at a local min:
point:
[ 1. 1.]
function value:
0.0
autodiff gradient:
[-0. 0.]
finite differences gradient:
[ 0. 0.]
autodiff hessian:
[[ 802. -400.]
[-400. 200.]]
finite differences hessian:
[[ 802. -400.]
[-400. 200.]]
---------------------------------------------------------
searches beginning from the easier init point [ 2. 2.]
---------------------------------------------------------
properties of the function at the initial guess:
point:
[ 2. 2.]
function value:
401.0
autodiff gradient:
[ 1602. -400.]
finite differences gradient:
[ 1602. -400.]
autodiff hessian:
[[ 4002. -800.]
[ -800. 200.]]
finite differences hessian:
[[ 4002. -800.]
[ -800. 200.]]
strategy: default (Nelder-Mead)
options: default
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 62
Function evaluations: 119
[ 0.99998292 0.99996512]
strategy: ncg
options: default
gradient: autodiff
hessian: autodiff
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 33
Function evaluations: 52
Gradient evaluations: 33
Hessian evaluations: 33
[ 0.99996674 0.99993334]
strategy: ncg
options: default
gradient: autodiff
hessian: finite differences
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 33
Function evaluations: 52
Gradient evaluations: 139
Hessian evaluations: 0
[ 0.99996668 0.99993322]
strategy: cg
options: default
gradient: autodiff
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 20
Function evaluations: 73
Gradient evaluations: 51
[ 0.99999877 0.99999753]
strategy: cg
options: default
gradient: finite differences
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 24
Function evaluations: 249
Gradient evaluations: 54
[ 0.99999552 0.99999104]
strategy: bfgs
options: default
gradient: autodiff
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 19
Function evaluations: 27
Gradient evaluations: 27
[ 0.99999999 0.99999999]
strategy: bfgs
options: default
gradient: finite differences
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 19
Function evaluations: 108
Gradient evaluations: 27
[ 0.99999551 0.99999102]
strategy: slsqp
options: default
gradient: autodiff
Optimization terminated successfully. (Exit mode 0)
Current function value: 1.0334670512e-07
Iterations: 20
Function evaluations: 29
Gradient evaluations: 20
[ 0.9996964 0.99938232]
strategy: slsqp
options: default
gradient: finite differences
Optimization terminated successfully. (Exit mode 0)
Current function value: 1.06959048795e-07
Iterations: 20
Function evaluations: 89
Gradient evaluations: 20
[ 0.99969095 0.9993713 ]
strategy: powell
options: default
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 12
Function evaluations: 339
[ 1. 1.]
strategy: tnc
options: default
gradient: autodiff
(array([ 1.00005974, 1.00011971]), 47, 1)
strategy: tnc
options: default
gradient: finite differences
(array([ 0.99999944, 0.99999889]), 64, 1)
---------------------------------------------------------
searches beginning from the more difficult init point [-1.2 1. ]
---------------------------------------------------------
properties of the function at the initial guess:
point:
[-1.2 1. ]
function value:
24.2
autodiff gradient:
[-215.6 -88. ]
finite differences gradient:
[-215.6 -88. ]
autodiff hessian:
[[ 1330. 480.]
[ 480. 200.]]
finite differences hessian:
[[ 1330. 480.]
[ 480. 200.]]
strategy: default (Nelder-Mead)
options: default
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 85
Function evaluations: 159
[ 1.00002202 1.00004222]
strategy: ncg
options: default
gradient: autodiff
hessian: autodiff
Optimization terminated successfully.
Current function value: 3.811010
Iterations: 39
Function evaluations: 41
Gradient evaluations: 39
Hessian evaluations: 39
[-0.95155681 0.91039596]
strategy: ncg
options: default
gradient: autodiff
hessian: finite differences
Optimization terminated successfully.
Current function value: 3.810996
Iterations: 39
Function evaluations: 41
Gradient evaluations: 185
Hessian evaluations: 0
[-0.95155309 0.91038895]
strategy: cg
options: default
gradient: autodiff
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 10
Function evaluations: 41
Gradient evaluations: 36
[ 1. 0.99999999]
strategy: cg
options: default
gradient: finite differences
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 13
Function evaluations: 221
Gradient evaluations: 51
[ 1.00000015 1.00000031]
strategy: bfgs
options: default
gradient: autodiff
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 31
Function evaluations: 45
Gradient evaluations: 45
[ 0.99999933 0.99999865]
strategy: bfgs
options: default
gradient: finite differences
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 31
Function evaluations: 180
Gradient evaluations: 45
[ 0.99999486 0.9999897 ]
strategy: slsqp
options: default
gradient: autodiff
Optimization terminated successfully. (Exit mode 0)
Current function value: 1.12238027858e-08
Iterations: 34
Function evaluations: 47
Gradient evaluations: 34
[ 0.99992192 0.99985101]
strategy: slsqp
options: default
gradient: finite differences
Optimization terminated successfully. (Exit mode 0)
Current function value: 1.20063082621e-08
Iterations: 34
Function evaluations: 149
Gradient evaluations: 34
[ 0.99991762 0.99984247]
strategy: powell
options: default
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 23
Function evaluations: 665
[ 1. 1.]
strategy: tnc
options: default
gradient: autodiff
(array([ 1.00000187, 1.00000376]), 76, 1)
strategy: tnc
options: default
gradient: finite differences
(array([ 0.99995432, 0.99990844]), 97, 1)
```

The best way to find the minimum of this function is not so clear. One confounding factor is that the various search strategies do not necessarily have comparable default stopping criteria. Another factor is that the direct function evaluation, the AlgoPy gradient evaluation, and the finite differences gradient approximation may each have different evaluation speeds.

On one hand the Newton conjugate gradient search fails to find the right minimum despite its wealth of AlgoPy-provided information about the objective function, whereas the Powell search finds the minimum using only direct evaluation of the objective function. On the other hand we see that the BFGS search, which succeeds in finding the minimum, has improved accuracy when it uses AlgoPy to compute the gradient as opposed to computing the gradient by finite differences.

Although not implemented here, perhaps the nonlinear optimization search strategies available in IPOPT would make better use of the gradient and hessian, as suggested by this vignette for the R interface to IPOPT. This comparison has not yet been made using AlgoPy, because the Python interface to IPOPT is more complicated to set up and use than the scipy.optimize procedures.