Introduction to Unconstrained Optimization - Scilab

derivative
numerical derivatives of a function.
Manages arbitrary step size.
Provides order 1, 2 or 4 finite difference formula.
Computes Jacobian and Hessian matrices.
Manages additionnal arguments of the function to derivate.
Produces various shapes of the output matrices.
Manages orthogonal basis change Q.
Figure 2: The derivative function.
when we develop an objective function and is derivatives. In order to help ourselves,
we can use the function values which were given previously. In the following session,
we check that our implementation of Rosenbrock’s function satisfies f(x 0 ) = 24.2
and f(x ⋆ ) = 0.
-->x0 = [ -1.2 1.0] ’
x0 =
- 1.2
1.
-->f0 = rosenbrock ( x0 )
f0 =
24.2
--> xopt = [1.0 1.0] ’
xopt =
1.
1.
--> fopt = rosenbrock ( xopt )
fopt =
0.
Notice that we define the initial and final points x0 and xopt as column vectors
(as opposed to row vectors). Indeed, this is the natural orientation for Scilab, since
this is in that order that Scilab stores the values internally (because Scilab was
primarily developped so that it can access to the Fortran routines of Lapack). Hence,
we will be consistently use that particular orientation in this document. In practice,
this allows to avoid orientation problems and related bugs.
We must now check that the derivatives are correctly implemented in our rosenbrock
function. To do so, we may use a symbolic computation system, such as Maple
[10], Mathematica [11] or Maxima [1]. An effective way of computing a symbolic
derivative is to use the web tool Wolfram Alpha [12], which can be convenient in
simple cases.
Another form of cross-checking can be done by computing numerical derivatives
and checking against the derivatives computed by the rosenbrock function. Scilab
provides one function for that purpose, the derivative function, which is presented
in figure 2. For most practical uses, the derivative function is extremely versatile,
and this is why we will use it thoughout this document.
In the following session, we check our exact derivatives against the numerical
derivatives produced by the derivative function. We can see that the results are
very close to each other, in terms of relative error.
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