Several signal recovery tasks can be relaxed into semidefinite programs with
rank-one minimizers. A common technique for proving these programs succeed is
to construct a dual certificate. Unfortunately, dual certificates may not exist
under some formulations of semidefinite programs. In order to put problems into
a form where dual certificate arguments are possible, it is important to
develop conditions under which the certificates exist. In this paper, we
provide an example where dual certificates do not exist. We then present a
completeness condition under which they are guaranteed to exist. For programs
that do not satisfy the completeness condition, we present a completion process
which produces an equivalent program that does satisfy the condition. The
important message of this paper is that dual certificates may not exist for
semidefinite programs that involve orthogonal measurements with respect to
positive-semidefinite matrices. Such measurements can interact with the
positive-semidefinite constraint in a way that implies additional linear
measurements. If these additional measurements are not included in the problem
formulation, then dual certificates may fail to exist. As an illustration, we
present a semidefinite relaxation for the task of finding the sparsest element
in a subspace. One formulation of this program does not admit dual
certificates. The completion process produces an equivalent formulation which
does admit dual certificates