The MOSEK Python optimizer API manual Version 7.0 (Revision 141)

10.3. SEMIDEFINITE OPTIMIZATION 131
10.3.2 Infeasibility for semidefinite optimization
In case MOSEK finds a problem to be infeasible it reports a certificate of the infeasibility. This works
exactly as for linear problems (see Section 10.1.2).
10.3.2.1 Primal infeasible problems
If the problem (10.10) is infeasible, MOSEK will report a certificate of primal infeasibility: The dual
solution reported is a certificate of infeasibility, and the primal solution is undefined.
A certificate of primal infeasibility is a feasible solution to the problem
maximize
subject to
(l c ) T s c l − (u c ) T s c u + (l x ) T s x l − (u x ) T s x u
A T y + s x l − s x u + s x n = 0,
m−1
∑
y i A ij + S j = 0, j = 0, . . . , p − 1
i=0
− y + s c l − s c u = 0,
s c l , s c u, s x l , s x u ≥ 0,
s x n ∈ C ∗ , S j ∈ S r + j
, j = 0, . . . , p − 1
such that the objective value is strictly positive.
(10.12)
10.3.2.2 Dual infeasible problems
If the problem (10.11) is infeasible, MOSEK will report a certificate of dual infeasibility: The primal
solution reported is the certificate of infeasibility, and the dual solution is undefined.
A certificate of dual infeasibility is a feasible solution to the problem
where
minimize
n−1
∑ ∑p−1
〈 〉
c j x j + Cj , X j
j=0
j=0
p−1
subject to ˆlc i ≤ ∑ ∑ 〈 〉
a ij x j + Aij , X j
j=1
ˆlx
j=0
≤ û c i, i = 0, . . . , m − 1
≤ x ≤ û x ,
x ∈ C, X j ∈ S + r j
, j = 0, . . . , p − 1
(10.13)
and
{ 0 if l
c
ˆlc i = i > −∞,
− ∞ otherwise,
{ 0 if l
x
ˆlx j = j > −∞,
− ∞ otherwise,
{
and û c 0 if u
c
i :=
i < ∞,
∞ otherwise,
{ 0 if u
and û x x
j :=
j < ∞,
∞ otherwise,