Nonlinear, multi-parametric programming with regularization

Question

I'm considering the following nonlinear, multiparametric problem with regularization term: $$ x(\theta,\sigma)=\underset{y\in\Omega}{\textrm{argmin}}~f(y,\theta)+\sigma\|y\|_2^2, $$ where $\Omega \subseteq \mathbb R^n$ is convex and nonempty, $y\mapsto f(y)$ is continuous and convex, $\theta\mapsto f(y,\theta)$ is continuous with $\theta \in \mathbb{R}^p$ and $\sigma>0$. Consider now a sequence for $\theta$ and $\sigma$ so that $\lim_{k\to\infty} \theta_k = \bar \theta$ and $\lim_{k\to\infty} \sigma_k = 0$. Can one infer some continuity property for the sequence $x(\theta_k,\sigma_k)$ w.r.t. the first argument as $k\to\infty$?