Introduction to Unconstrained Optimization - Scilab

1.5
1.0
8
12
0.5
0.0
-0.5
0.5
2
-1.0
-1.5
-2.0
8
12
8
-2.5
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
Figure 27: The contours of a quadratic function.
In order to check that the computations are correct, we use the derivative function.
-->[gfd , Hfd ]= derivative ( quadratic , x, H_form =’ blockmat ’)
Hfd =
2. 1.
1. 4.
gfd =
4. 7.
We finally compute the relative error between the computed gradient and Hessian and the finite
difference formulas.
--> norm (g-gfd ’)/ norm (g)
ans =
3.435D -12
--> norm (H-Hfd )/ norm (H)
ans =
0.
The relative error for the gradient indicates that there are approximately 12 significant digits.
Therefore, our gradient is accurate. The Hessian matrix is exact.
3.2 Answers for section 2.8
Answer of Exercise 2.1 (Convex hull - 1 )
Before really detailing the proof, we can detail an auxiliary result, which will help us in the
design of the proof. We are going to prove that a convex combination of 2 points can be combined
with a third point so that the result is a convex combination of the 3 points. Let us suppose that
C is a convex set et let us assume that three points x 1 , x 2 , x 3 are in C. Let us assume that x 2 is
a convex combination of x 1 and x 2 , i.e.
x 2 = θ 2 x 1 + (1 − θ 2 )x 2 , (64)
with 0 ≤ θ 2 ≤ 1. Let us define x 3 as a convex combination of x 2 and x 3 , i.e.
x 3 = θ 3 x 2 + (1 − θ 3 )x 3 , (65)
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