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

132 CHAPTER 10. PROBLEM FORMULATION AND SOLUTIONS
such that the objective value is strictly negative.
10.4 Quadratic and quadratically constrained optimization
A convex quadratic and quadratically constrained optimization problem is an optimization problem of
the form
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 kj x j ≤ u c k, k = 0, . . . , m − 1,
j=0
l x j ≤ x j ≤ u x j , j = 0, . . . , n − 1,
(10.14)
where Q o and all Q k are symmetric matrices. Moreover for convexity, Q o must be a positive semidefinite
matrix and Q k must satisfy
− ∞ < lk c ⇒ Q k is negative semidefinite,
u c k < ∞ ⇒ Q k is positive semidefinite,
− ∞ < lk c ≤ u c k < ∞ ⇒ Q k = 0.
The convexity requirement is very important and it is strongly recommended that MOSEK is applied
to convex problems only.
Note that any convex quadratic and quadratically constrained optimization problem can be reformulated
as a conic optimization problem. It is our experience that for the majority of practical applications
it is better to cast them as conic problems because
• the resulting problem is convex by construction, and
• the conic optimizer is more efficient than the optimizer for general quadratic problems.
See [7] for further details.
10.4.1 Duality for quadratic and quadratically constrained optimization
The dual problem corresponding to the quadratic and quadratically constrained optimization problem
(10.14) is given by
maximize
(
(l c ) T s c l − (u c ) T s c u + (l x ) T s x l − (u x ) T s x u + 1 m−1 ∑
2 xT k=0
y k Q k − Q o )
x + c f
subject to A T y + s x l − s x u +
( m−1
) ∑
y k Q k − Q o x
k=0
= c,
− y + s c l − s c u = 0,
s c l , s c u, s x l , s x u ≥ 0.
(10.15)