Consider the unconstrained optimization problem

whereare given. The goal of this exercise is to determine the optimal value p and the set of optimal solutions, ,in terms of c and the eigenvalues and eigenvectors of the (symmetric) matrix Q.

1. Assume that Show that the optimal set is a singleton, and that p is finite. Determine both in terms of Q, c.

2.

3. Now we do not assume that Q is diagonal anymore. Under what conditions (on Q, c)) is the optimal value finite? Make sure to express your result in terms of Q and c, as explicitly as possible.

Now we do not assume that Q is diagonal anymore. Under what
conditions (on Q, c) is the optimal value finite? Make sure to express
your result in terms of Q and c, as explicitly as possible.
4. Assuming that the optimal value is finite, determine the optimal value and optimal set. Be as specific as you can, and express your results in terms of the pseudo-inverse⁵ of Q.

Transcribed Image Text:

p* T 1 1/2 x ¹ Qx - c¹ x = min x