Introduction to Unconstrained Optimization - Scilab

negative eigenvalues.) Assume that b = 0 and the Hessian matrix is
( ) 3 −1
H =
−1 −8
(58)
The eigenvalues are approximately equal to (λ 1 , λ 2 ) ≈ (−8.0901699, 3.0901699) so
that the matrix is indefinite. Hence, this quadratic function is neither convex, nor
concave. The following script produces the contours of the corresponding quadratic
function.
function f = quadraticsaddle ( x1 , x2 )
x = [x1 x2]’
H = [3 -1; -1 -8]
f = x.’ * H * x;
endfunction
x = linspace ( -2 ,2 ,100);
y = linspace ( -2 ,2 ,100);
contour ( x , y , quadraticsaddle , ..
[ -20 -10 -5 -0.3 0.3 2 5 10])
The contour plot is presented in figure 23.
The following script allows to produce the 3D plot presented in figure 24.
function f = quadraticsaddle ( x1 , x2 )
x = [x1 ’ x2 ’] ’
H = [3 -1; -1 -8]
y = H * x
n = size (y,"c")
for i = 1 : n
f(i) = x(: ,i)’ * y(: ,i)
end
endfunction
x = linspace ( -2 ,2 ,20);
y = linspace ( -2 ,2 ,20);
Z = ( eval3d ( quadraticsaddle ,x,y)) ’;
surf (x,y,Z)
h = gcf ();
cmap = graycolormap (10)
h. color_map = cmap ;
The contour is typical of a saddle point where the gradient is zero, but which is
not a local minimum. Notice that the function is unbounded.
Example 2.4 (A quadratic function for which the Hessian has a zero and a positive
eigenvalue.) Assume that b = 0 and the Hessian matrix is
( ) 4 2
H =
(59)
2 1
The eigenvalues are approximately equal to (λ 1 , λ 2 ) ≈ (0, 5) so that the matrix is
indefinite. Hence, this quadratic function is convex (but not strictly convex). The
following script produces the contours of the corresponding quadratic function which
are presented in the figure 25.
function f = quadraticindef ( x1 , x2 )
x = [x1 x2]’
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