Sensitivity analysis in conic optimization I have a conic optimization of the form:
$$\min_x \langle c, x \rangle,\ \text{s.t.}\ Ax = b,\ x \in K.$$
where $x \in \mathbb{R}^{n}$, $A$ is an $m \times n$ matrix, $b \in \mathbb{R}^m$, $K$ is a self dual cone in $\mathbb{R}^n$ and $\langle~,~\rangle$ is the standard Euclidean inner product on $\mathbb{R}^n$.
I am looking for conditions under which the optimal value function is a continuous function of perturbations in the vector $b$.  In particular, if we replace $b$ with $b + \Delta b$, so the linear constraint becomes $Ax = b + \Delta b$, and if we let $\phi ( \Delta b)$ denote the optimal value of this perturbed problem, I am interested in when $| \phi(0) - \phi(\Delta b) | \rightarrow 0$ as $\| \Delta b\|_{\infty} \rightarrow 0$.
I have seen results that show $\phi(\Delta b)$ is a linear function in $\Delta b$ for perturbations that preserve the optimal partitions, but I am interested in the case when the optimal partition is not necessarily preserved.  
 A: In general, your problem can become infeasible under arbitrarily small perturbations $\Delta b$.  For example, consider the problem 
$\min X_{11}+X_{22}$
subject to 
$X_{11}=0$
$X_{22}=0$
$X \succeq 0$.
The only feasible solution $X=0$, and the problem becomes infeasible if $\Delta b_{1}$ or $\Delta b_{2}$ is negative.  
The book "Perturbation Analysis of Optimization Problems" by Bonnans and Shapiro has some relevant theorems, but as I recall, there's nothing really useful.  
A: I'm not sure, but the proof of Proposition 2.3 in Ekeland and Temam might be relevant.  I'll summarize / quote from Ekeland and Temam:
Assume that $\Phi$ is a closed, convex, proper function on $V \times Y$, where $V$ and $Y$ are (Hausdorff) topological vector spaces.  (Perhaps for simplicity we should just assume $V$ and $Y$ are finite dimensional inner product spaces over $\mathbb R$.)  For $p \in Y$ let 
\begin{equation}
h(p) = \inf_{u \in V} \Phi(u,p).
\end{equation}
(So $h$ is the optimal value function.  The unperturbed primal problem is to minimize $\Phi(u,0)$ with respect to $u$.)
Assume that $\inf_u \Phi(u,0)$ is finite and that
\begin{equation}
\text{There exists $u_0 \in V$ such that $p \mapsto \Phi(u_0,p)$ is finite
and continuous at $0 \in Y$.}
\end{equation}
(This assumption can be compared with Slater's condition.)  
The proof of Proposition 2.3 argues as follows:
Note that $h$ is convex and $h(0)$ is finite.
As the function $p \mapsto \Phi(u_0,p)$ is convex and continuous at $0 \in Y$,
there exists a neighborhood $\mathcal V$ of $0$ in $Y$, on which this function is bounded above:
\begin{equation}
\Phi(u_0,p) \leq M < +\infty, \quad \forall p \in \mathcal V.
\end{equation}
But
\begin{equation}
h(p) = \inf_{u \in V} \Phi(u,p) \leq \Phi(u_0,p) \leq M, \quad \forall p \in \mathcal V
\end{equation}
from which it follows (by proposition I.2.5 in Ekeland and Temam) that $h$ is continuous at $0$.
(Corollary I.2.5 states that every closed convex function on a Banach space is continuous over the interior of its effective domain.)
(Most of what I've written is taken word from word from Ekeland and Temam.)
A: This bibliography should help:
"Semidefinite and Cone Programming Bibliography/Comments", Henry Wolkowicz, 2011 (http://www.math.uwaterloo.ca/~hwolkowi/henry/software/sdpbibliog.pdf)
...or have you settled the question in the meantime?
