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

32 CHAPTER 7. A GUIDED TOUR
1 % cqo2.m
2
3 [r, res] = mosekopt(’symbcon’);
4
5 % Set up the non-conic part of the problem.
6 prob = [];
7 prob.c = [1 1 1 0 0 0 0]’;
8 prob.a = sparse([[1 0 1 0 1 0 0];...
9 [0 1 0 0 0 0 -1]]);
10 prob.blc = [0.5 0];
11 prob.buc = [0.5 0];
12 prob.blx = [-inf -inf -inf 1 -inf 1 -inf];
13 prob.bux = [inf inf inf 1 inf 1 inf];
14
15 % Set up the cone information.
16 prob.cones.type = [res.symbcon.MSK CT QUAD, ...
17 res.symbcon.MSK CT RQUAD];
18 prob.cones.sub = [4, 1, 2, 3, 5, 6, 7];
19 prob.cones.subptr = [1, 5];
20
21 [r,res] = mosekopt(’minimize’,prob);
22
23 % Display the solution.
24 res.sol.itr.xx’
[ cqo2.m ]
7.6.4 Conic duality and the dual solution
The dual problem corresponding to the conic optimization problem (7.5) is given by
maximize (l c ) T s c l − (u c ) T s c u
+ (l x ) T s x l − (u x ) T s x u + c f
subject to − y + s c l − s c u = 0,
A T y + s x l − s x u + s x n = c,
s c l , s c u, s x l , s x u ≥ 0,
s x n ∈ C ∗
where the dual cone C ∗ is defined as follows. Let (s x n) be partitioned similar to x, i.e. if x j is a member
of x t , then (s x n) j is a member of (s x n) t as well. Now, the dual cone is defined by
C ∗ :=
{
}
s x n ∈ R nt
: (s x n) t ∈ Ct ∗ , t = 1, . . . , k
where the type of Ct
∗ is dependent on the type of C t . For the cone types MOSEK can handle the
relation between the primal and dual cones is given as follows:
• R set:
C t =
{
x ∈ R nt} {
}
⇔ Ct ∗ := s ∈ R nt : s = 0 .