\subsection{Basic Duality and Constraint Qualifications}
The approach (as suggested by Shapiro) will be for general cone problems
and using the 'Rockafellar conjugate duality' approach.

However, I think we need to have the basic SDP results first.

I enter the basic results - we need to fill in the blanks:

\begin{enumerate}

\item
basic duality SDP pair - derivation using min-max approach - we get weak
duality and feasibility from unboundedness conclusions.

This includes Lagrangian, polar cone, adjoint operator, ...

Then generalize to cone programs.


\item
Now discuss strong duality - the need for CQs - the weakest CQs in terms
of the closure of two cones - difference with LP case

include generic property of strict feasibility

\item
complementary slackness and strict complementary slackness including
'generic property'  - difference with LP case

\item
One then needs to discuss the need for CQs and different CQs and results
without CQs. - here Ramana et al, and Borwein et al.

There are many constraint qualifications that can work for SDP, e.g. the
best known is:
   Slater's CQ.
Different optimality conditions 
with and without a CQ are given in  e.g.
\cite{bw4,bw3,bw2,bw1,w11}
\cite{MR96e:15039,MR1430294}
\cite{BoWo:86,Ram:95,RaTuWo:95}.

For example, in \cite{Lass:96}, there is a CQ which does not require
Slater's condition, i.e. one only require
${\cal A} X \succeq C, X \succeq 0$ have a finite optimum and $C \succ
0.$ In addition, the paper discusses a Farkas' lemma, existence of
solutions, and geometry of BFS.
Note that this CQ is implied by the one:   the optimal set is a bounded
set. This latter is equivalent to the dual satisfies Slater's. So this
CQ is NOT an extension from Slater's.



\item
Here we discuss just duality gaps as opposed to strong duality. There
are conditions that guarantee no duality gap - though dual attainment
may fail.

There are several papers by Rockafellar which are relevant:
e.g. \cite{Rocka2:71} contains the result that faithfully convex programs
do not have a duality gap - this includes quadratic-quadratic convex
programs; \cite{Rocka2:65} contains results on perturbations of convex
inequalities and so again has applications to certain SDPs, e.g. as a
corollary we get a family of SDP's with no duality gap but no Slater
condition required.

\item
Now we work on optimality conditions as opposed to dual programs. These
are closely tied but can differ, e.g. in the case of no dual attainment,
the dual program essentially works but the optimality conditions fail.

Here we should do first and second order optimality conditions a la
Shapiro ...


\end{enumerate}


\subsection{Sensitivity Analysis}

\subsection{Parametric SDP}
\subsection{Degeneracy }