The MOSEK command line tool Version 7.0 (Revision 141)

4.5. GENERAL CONVEX OPTIMIZATION 31
L(x, s c l , s c u, s x l , s x u) := f(x) + c T x + c f
and the dual problem is given by
− (s c l ) T (g(x) + Ax − l c ) − (s c u) T (u c − g(x) − Ax)
− (s x l ) T (x − l x ) − (s x u) T (u x − x),
which is equivalent to
maximize L(x, s c l , s c u, s x l , s x u)
subject to ∇ x L(x, s c l , s c u, s x l , s x u) T = 0,
s c l , s c u, s x l , s x u ≥ 0,
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 + c f
+ f(x) − g(x) T y − (∇f(x) T − ∇g(x) T y) T x
subject to A T y + s x l − s x u − (∇f(x) T − ∇g(x) T y) = c,
− y + s c l − s c u = 0,
s c l , s c u, s x l , s x u ≥ 0.
In this context we use the following definition for scalar functions
and accordingly for vector functions
∇f(x) =
∇g(x) =
[ ∂f(x)
∂x 1
, ...,
⎡
⎣
∇g 1 (x)
:
∇g m (x)
]
∂f(x)
,
∂x n
⎤
⎦ .
(4.18)