Introduction to Unconstrained Optimization - Scilab

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exp(x)
x^2
100
50
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
3.5
3.0
-log(x)
x*log(x)
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Figure 19: Examples of convex functions.
x = linspace ( -5 , 5 , 1000 );
plot ( x , exp (x) , "b-" )
plot ( x , x^2 , "g--" )
legend ( [ " exp (x)" "x^2"] )
//
x = linspace ( 0.1 , 3 , 1000 );
plot ( x , -log (x) , "b-" )
plot ( x , x.* log (x) , "g--" )
legend ( [ "-log (x)" "x* log (x)"] )
As seen in the proof of the proposition 2.7, the hypothesis that the domain of
definition of f, denoted by C, is convex cannot be dropped. For example, consider
the function f(x) = 1/x 2 , defined on the set C = {x ∈ R, x ≠ 0}. The second
derivative of f is positive, since f ′′ (x) = 6/x 4 > 0. But f is not a convex function
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