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

130 CHAPTER 10. PROBLEM FORMULATION AND SOLUTIONS
10.3 Semidefinite optimization
Semidefinite optimization is an extension of conic quadratic optimization (see Section 10.2) allowing
positive semidefinite matrix variables to be used in addition to the usual scalar variables. A semidefinite
optimization problem can be written as
minimize
subject to l c i ≤
n−1
∑ ∑p−1
〈 〉
c j x j + Cj , X j + c
f
j=0
n−1
j=0
p−1
∑ ∑ 〈 〉
a ij x j + Aij , X j
j=0
j=0
≤ u c i, i = 0, . . . , m − 1
lj x ≤ x j ≤ u x j , j = 0, . . . , n − 1
x ∈ C, X j ∈ S r + j
, j = 0, . . . , p − 1
(10.10)
where the problem has p symmetric positive semidefinite variables X j ∈ S r + j
of dimension r j with
symmetric coefficient matrices C j ∈ S rj and A i,j ∈ S rj . We use standard notation for the matrix inner
product, i.e., for U, V ∈ R m×n we have
〈U, V 〉 :=
m−1
∑
i=0
n−1
∑
U ij V ij .
With semidefinite optimization we can model a wide range of problems as demonstrated in [7].
j=0
10.3.1 Duality for semidefinite optimization
The dual problem corresponding to the semidefinite optimization problem (10.10) is given by
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 + c f
c − A T y + s x u − s x l = s x n,
m∑
C j − y i A ij = S j , j = 0, . . . , p − 1
i=0
s c l − s c u = y,
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
(10.11)
where A ∈ R m×n , A ij = a ij , which is similar to the dual problem for conic quadratic optimization (see
Section 10.7), except for the addition of dual constraints
m∑
(C j − y i A ij ) ∈ S r + j
.
Note that the dual of the dual problem is identical to the original primal problem.
i=0