The MOSEK optimization toolbox for MATLAB manual Version 7.0 (Revision 141)

26 CHAPTER 7. A GUIDED TOUR
which for the example (7.3) implies that
and that
⎡
Q = ⎣
1
minimize
2 xT Qx + c T x
subject to l c ≤ Ax, ≤ u c ,
l x ≤ x ≤ u x ,
2 0 − 1
0 0.2 0
− 1 0 2
⎤
⎡
⎦ , c = ⎣
⎡
l c = 1, u c = ∞, l x = ⎣
0
0
0
⎤
0
− 1
0
⎤
⎦ , A = [ 1 1 1 ] ,
⎡
⎦ and u x = ⎣
∞
∞
∞
⎤
⎦
(7.4)
Please note the explicit 1 2
in the objective function of (7.4) which implies that diagonal elements must
be doubled in Q, i.e. Q 11 = 2, whereas the coefficient in (7.3) is 1 in front of x 2 1.
7.5.1 Two important assumptions
MOSEK assumes that the Q matrix is symmetric, i.e.
Q = Q T
and that Q is positive semidefinite.A matrix is positive semidefinite if the smallest eigenvalue of the
matrix is nonnegative. An alternative statement of the positive semidefinite requirement is
x T Qx ≥ 0, ∀x.
If Q is not positive semidefinite, then MOSEK will not produce reliable results or work at all.
One way of checking whether Q is positive semidefinite is to check whether all the eigenvalues of Q
are nonnegative. The MATLAB command eig computes all eigenvalues of a matrix.
7.5.2 Using mskqpopt
The subsequent MATLAB statements solve the problem (7.3) using the mskqpoptMOSEK function
1 % qo1.m
2
3 % Set up Q.
4 q = [[2 0 -1];[0 0.2 0];[-1 0 2]];
5
6 % Set up the linear part of the problem.
7 c = [0 -1 0]’;
[ qo1.m ]