By a theorem of Goldman and Tucker it is known that if a linear program (LP) has a finite valued optimal solution, then there is an optimal primal/dual pair $(x,z)$ satisfying not only *complementary slackness* (i.e. $x_iz_i=0$), but also *strict complementary slackness* (i.e. *exactly one* of $x_i$ and $z_i$ vanishs).

It is known that this property does not carry over to semidefinite programs (SDP). This means, if $(X,Z)$ is an optimal primal/dual pair of an SDP, where $X,Z\in\Bbb R^{n\times n}$ are symmetric positive semi-definite matrices, then we have complementarity (i.e. $XZ=0$), but in general no strict complementarity $\def\rank{\mathrm{rank}}\rank(X)+\rank(Z)=n$.

Now, I am especially interested in **SDPs with finite valued optimal solutions and zero duality gap**. Is there anything more we can say about strict complementary slackness in the case of strong duality? I actually have never seen an example where it fails, i.e. an example with above properties where all optimal primal/dual pairs $(X,Z)$ lack strict complementarity.

Most of my knowledge about strict complementarity comes from [1] where it is shown that it is a generic property (holds for almost all SDPs in a precise sense). Besides this, strict complementarity was mostly assumed to prove other properties.