Introduction to Unconstrained Optimization - Scilab

2.0
1.5
20
1.0
0.5
10
0.0
5
2
0.3 0.3 2
5
-0.5
10
-1.0
20
-1.5
-2.0
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
Figure 25: Contour of a quadratic function – One eigenvalue is zero, the other is
positive.
H = [4 2; 2 1]
f = x.’ * H * x;
endfunction
x = linspace ( -2 ,2 ,100);
y = linspace ( -2 ,2 ,100);
contour ( x , y , quadraticindef , [0.3 2 5 10 20])
The contour is typical of a weak local optimum. Notice that the function remains
constant along the eigenvector corresponding with the zero eigenvalue.
Example 2.5 (Example where there is no stationnary point.) Assume that b =
(1, 0) T and the Hessian matrix is
( ) 0 0
H =
(60)
0 1
The function can be simplified as f(x) = x 1 + 1 2 x2 2. The gradient of the function
is g(x) = (1, x 2 ) T . There is no stationnary point, which implies that the function
is unbounded. The following script produces the contours of the corresponding
quadratic function which are presented in the figure 26.
function f = quadraticincomp ( x1 , x2 )
x = [x1 x2]’
H = [0 0;0 1]
b = [1;0]
f = x.’*b + x.’ * H * x;
endfunction
x = linspace ( -10 ,10 ,100);
y = linspace ( -10 ,10 ,100);
contour ( x , y , quadraticincomp , [ -10 -5 0 5 10 20])
32