Introduction to Unconstrained Optimization - Scilab

often in an optimization context, and particularily in this document. Still, for more
than two parameters, no contour plot can be drawn. Therefore, when the objective
function depends on more than two parameters, it is not easy to represent ourselves
its associated landscape.
Although this is not an issue in most cases, it might happen that the performance
might be a problem to create the contour plot. In this case, we can rely on the
vectorization features of Scilab, which allows to greatly improve the speed of the
creation of the computations.
In the following script, we create a rosenbrockV function which is a vectorized
version of the Rosenbrock function. It takes the m-by-n matrices x1 and x2 as input
arguments and returns the m-by-n matrix f which contains all the required function
values. We use of the elementwise operators .* and .^, which apply to a matrix of
values element-by-element operations. Then we use the meshgrid function, which
creates the two 100-by-100 matrices XX1 and XX2, containing combinations of the
1-by-100 matrices x1 and x2. This allows to compute the 100-by-100 matrix Z in
one single call to the rosenbrockV function. Finally, we call the contour function
which creates the same figure as previously.
function f = rosenbrockV ( x1 , x2 )
f = 100 *(x2 -x1 .^2).^2 + (1-x1 ).^2
endfunction
x1 = linspace ( -2 ,2 ,100);
x2 = linspace ( -2 ,2 ,100);
[XX1 , XX2 ]= meshgrid (x1 ,x2 );
Z = rosenbrockV ( XX1 , XX2 );
contour ( x1 , x2 , Z’ , [1 10 100 500 1000] );
The previous script is significantly faster than the previous versions. But it requires
to rewrite the Rosenbrock function as a vectorized function, which can be impossible
in some practical cases.
1.5 What is an optimum ?
In this section, we present the characteristics of the solution of an optimization
problem. The definitions presented in this section are mainly of theoretical value,
and, in some sense, of didactic value as well. More practical results will be presented
later.
For any given feasible point x ∈ R n , let us define N(x, δ) the set of feasible points
contained in a δ− neighbourhood of x.
A δ− neighbourhood of x is the ball centered at x with radius δ,
N(x, δ) = {y ∈ R n , ‖x − y‖ ≤ δ} . (16)
There are three different types of minimum :
• strong local minimum,
• weak local minimum,
• strong global minimum.
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