Conic linear programs, among them semidefinite programs, often behave
pathologically: the optimal values of the primal and dual programs may differ,
and may not be attained. We present a novel analysis of these pathological
behaviors. We call a conic linear system Ax<=b {\em badly behaved} if the
value of sup∣Ax<=b is finite but the dual program has no
solution with the same value for {\em some} c. We describe simple and
intuitive geometric characterizations of badly behaved conic linear systems.
Our main motivation is the striking similarity of badly behaved semidefinite
systems in the literature; we characterize such systems by certain {\em
excluded matrices}, which are easy to spot in all published examples.
We show how to transform semidefinite systems into a canonical form, which
allows us to easily verify whether they are badly behaved. We prove several
other structural results about badly behaved semidefinite systems; for example,
we show that they are in NP∩co−NP in the real number model of computing.
As a byproduct, we prove that all linear maps that act on symmetric matrices
can be brought into a canonical form; this canonical form allows us to easily
check whether the image of the semidefinite cone under the given linear map is
closed.Comment: For some reason, the intended changes between versions 4 and 5 did
not take effect, so versions 4 and 5 are the same. So version 6 is the final
version. The only difference between version 4 and version 6 is that 2 typos
were fixed: in the last displayed formula on page 6, "7" was replaced by "1";
and in the 4th displayed formula on page 12 "A_1 - A_2 - A_3" was replaced by
"A_3 - A_2 - A_1