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 ).