Given a positive integer vector 
,
with 
,
let 
denote the space of all real,
symmetric, 
block diagonal matrices whose 
diagonal block is of size 
.
The inner product on this space is
By 
, where 
, we mean that X is positive
semidefinite, i.e. all its diagonal blocks are positive semidefinite.
Consider the semidefinite program (SDP)
where C and 
, 
are all fixed matrices in 
,
and the unknown variable X also lies in 
. The dual program is
where the dual slack matrix Z also lies in 
.
In the special case 
, 
, the SDP reduces to a linear program.
It is assumed that the matrices 
, 
, are linearly
independent.
We shall use the notation
where 
denotes the Frobenius matrix norm.
Assuming a Slater condition,
i.e. the existence of a strictly feasible primal or dual point of SDP,
it is well known that the optimality conditions of SDP may be
expressed by the equations 
,
, and 
(together with the
semidefinite conditions 
and 
).