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

44 CHAPTER 5. BASIC API TUTORIAL
The second argument specify the semidefinite variable index j; in this example there is only a single
variable, so the index is 0. The next three arguments give the number of matrices used in the linear
combination, their indices (as returned by Task.appendsparsesymmat), and the weights for the individual
matrices, respectively. In this example, we form the objective matrix coefficient directly from a
single symmetric matrix.
Similary, a constraint matrix coefficient A ij is setup by the function Task.putbaraij:
123 task.putbaraij(0, 0, [syma0], [1.0])
124 task.putbaraij(1, 0, [syma1], [1.0])
[ sdo1.py ]
where the second argument specifies the constraint number (the corresponding row of A), and the
third argument specifies the semidefinite variable index (the corresponding column of A). The next
three arguments specify a weighted combination of symmetric matrices used to form the constraint
matrix coefficient.
After the problem is solved, we read the solution using Task.getbarxj:
146 task.getbarxj(mosek.soltype.itr, 0, barx)
[ sdo1.py ]
The function returns the half-vectorization of x j (the lower triangular part stacked as a column vector),
where the semidefinite variable index j is given in the second argument, and the third argument is a
pointer to an array for storing the numerical values.
5.5 Quadratic optimization
MOSEK can solve quadratic and quadratically constrained convex problems. This class of problems
can be formulated as follows:
1
minimize
2 xT Q o x + c T x + c f
subject to lk c ≤ 1 n−1
∑
2 xT Q k x + a k,j x j ≤ u c k, k = 0, . . . , m − 1,
j=0
l x j ≤ x j ≤ u x j , j = 0, . . . , n − 1.
(5.7)
Without loss of generality it is assumed that Q o and Q k are all symmetric because
x T Qx = 0.5x T (Q + Q T )x.
This implies that a non-symmetric Q can be replaced by the symmetric matrix 1 2 (Q + QT ).
The problem is required to be convex. More precisely, the matrix Q o must be positive semi-definite
and the kth constraint must be of the form