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

5.4. SEMIDEFINITE OPTIMIZATION 39
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
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 A, B ∈ R m×n we have
〈A, B〉 :=
m−1
∑
i=0
n−1
∑
A ij B ij .
j=0
5.4.1 Example: Semidefinite optimization
The problem
minimize
subject to
〈 ⎡ ⎤
2 1 0
〉
⎣ 1 2 1 ⎦ , X + x 0
0 1 2
〈 ⎡ ⎤
1 0 0
〉
⎣ 0 1 0 ⎦ , X + x 0 = 1,
0 0 1
〈 ⎡ ⎣ 1 1 1 ⎤ 〉
1 1 1 ⎦ , X + x 1 + x 2 = 1/2,
1
√
1 1
x 0 ≥ x 2 1 + x2 2 ,
X ≽ 0,
is a mixed semidefinite and conic quadratic programming problem with a 3-dimensional semidefinite
variable
⎡
X = ⎣
x 10
x 20
x 11
x 21
x 21
x 22
⎦ ∈ S 3 + ,
x 00 x 10 x 20
⎤
and a conic quadratic variable (x 0 , x 1 , x 2 ) ∈ Q 3 . The objective is to minimize
(5.6)
subject to the two linear constraints
2(x 00 + x 10 + x 11 + x 21 + x 22 ) + x 0 ,