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

5.5. QUADRATIC OPTIMIZATION 45
with a negative semi-definite Q k or of the form
lk c ≤ 1 n−1
∑
2 xT Q k x + a k,j x j (5.8)
j=0
n−1
1 ∑
2 xT Q k x + a k,j x j ≤ u c k. (5.9)
j=0
with a positive semi-definite Q k . This implies that quadratic equalities are not allowed. Specifying a
non-convex problem will result in an error when the optimizer is called.
5.5.1 Example: Quadratic objective
The following is an example of a quadratic, linearly constrained problem:
This can be written equivalently as
minimize x 2 1 + 0.1x 2 2 + x 2 3 − x 1 x 3 − x 2
subject to 1 ≤ x 1 + x 2 + x 3
x ≥ 0
where
Q o =
⎡
⎣ 0 0.2 0
2 0 − 1
− 1 0 2
minimize 1/2x T Q o x + c T x
subject to Ax ≥ b
x ≥ 0,
⎤
⎦ , c =
⎡
⎣ 0 − 1
0
⎤
⎦ , A = [ 1 1 1 ] , and b = 1.
Please note that MOSEK always assumes that there is a 1/2 in front of the x T Qx term in the objective.
Therefore, the 1 in front of x 2 0 becomes 2 in Q, i.e. Q o 0,0 = 2.
5.5.1.1 Source code
1 ##
[ qo1.py ]
2 # Copyright: Copyright (c) MOSEK ApS, Denmark. All rights reserved.
3 #
4 # File: qo1.py
5 #
6 # Purpose: Demonstrate how to solve a quadratic
7 # optimization problem using the MOSEK Python API.
8 ##
9