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.