Introduction to Unconstrained Optimization - Scilab

Exercise 2.5 (Convex function - 3 ) This exercise is associated with the proposition 2.6, which
gives the second form of the first order condition of a differentiable convex function. The first part
of the proof has already be proved in this chapter and the exercise is based on the proof of the
second part. Let f be continously differentiable on the convex set C. Prove that if f satisfies the
inequality
for all x, y ∈ C, then f is convex over the convex set C.
(g(x) − g(y)) T (x − y) ≥ 0 (61)
Exercise 2.6 (Hessian of Rosenbrock’s function) We have seen in exercise 1.1 that the
Hessian matrix of a quadratic function f(x) = b T x + 1 2 xT Hx is simply the matrix H. In this
case the proposition 2.7 states that if the matrix H is positive definite, then it is convex. On
the other hand, for a general function, there might be some points where the Hessian matrix is
positive definite and some other points where it is indefinite. Hence the Hessian positivity is only
local, as opposed to the global behaviour of a quadratic function. Consider Rosenbrock’s function
defined by the equation 10. Use Scilab and prove that the Hessian matrix is positive definite at
the point x = (1, 1) T . Check that it is indefinite at the point x = (0, 1) T . Make a random walk in
the interval [−2, 2] × [−1, 2] and check that many points are associated with an indefinite Hessian
matrix.
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