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

28 CHAPTER 7. A GUIDED TOUR
34
35 % Upper bounds of variables.
36 prob.bux = []; % There are no bounds.
37
38 [r,res] = mosekopt(’minimize’,prob);
39
40 % Display return code.
41 fprintf(’Return code: %d\n’,r);
42
43 % Display primal solution for the constraints.
44 res.sol.itr.xc’
45
46 % Display primal solution for the variables.
47 res.sol.itr.xx’
This sequence of commands looks much like the one that was used to solve the linear optimization
example using mosekopt except that the definition of the Q matrix in prob. mosekopt requires that
Q is specified in a sparse format. Indeed the vectors qosubi, qosubj, and qoval are used to specify
the coefficients of Q in the objective using the principle
Q qosubi(t),qosubj(t) = qoval(t), for t = 1, . . . , length(qosubi).
An important observation is that due to Q being symmetric, only the lower triangular part of Q should
be specified.
7.6 Conic optimization
One way of generalizing a linear optimization problem is to include a constraint of the form
x ∈ C
in the problem definition where C is required to be a convex cone. The resulting class of problems is
known as conic optimization. MOSEK can solve a subset of all conic problems and subsequently it is
demonstrated how to solve this subset using the mosekopt toolbox function.
7.6.1 The conic optimization problem
A conic optimization problem has the following form
minimize
c T x + c f
subject to l c ≤ Ax ≤ u c ,
l x ≤ x ≤ u x ,
x ∈ C,
where C must satisfy the following requirements. Let
(7.5)
x t ∈ R nt , t = 1, . . . , k