Exam 2

Exercise 2. Consider the following optimization problem:

$$\left\lbrace \begin{array}{ll} \mathrm{minimize} & 2x_1^2 +x_1 x_3 +x_2^2 +2x_2 x_3 +3x_3^2 +x_3 x_4 +2x_4^2 -5x_1 -4x_3 +3x_4 \\ x;\epsilon ;\Re^4 & ; \end{array}\right.$$

2.1 Do global optimal solutions exist? Why?

2.2 Is it a convex problem? Why?

2.3 Find the global minimum by using the gradient method with exact line search.

Starting from the point $\left(0,0,0,0\right);;\mathrm{with};\left|\nabla f\left(x\right)\right|<{10}^{-6}$ as stopping criterion. How many iterations are needed?

2.4 Find the global minimum by using the conjugate gradient method

Starting from the point (0, 0, 0, 0). How many iterations are needed? Write the point found by the method at any iteration.

2.5 Find the global minimum by using the Newton method

Starting from the point (0, 0, 0, 0). How many iterations are needed?