I am working on a research project that is related to optimization and recently had an idea to use MCMC in this setting. Unfortunately, I am fairly new to MCMC methods so I had several questions. I'll start by describing the problem and then asking my questions.
Our problem boils down to estimating the expected value of a cost function $c(\omega)$ where $\omega = (\omega_1,\omega_2,...\omega_h)$ is an $h$-dimentional random variable with a density $f(\omega)$.
In our case, a closed form version of $c(\omega)$ does not exist. This means that we have to use Monte Carlo methods to approximate the expected value. Unfortunately, it turns out that estimates of $E[c(\omega)]$ that are generated using MC or QMC methods have too much variance to be useful within in a practical setting.
One idea that we had to use an importance sampling distribution to generate sample points that will produce a low variance estimate of $E[c(\omega)]$. In our case, the ideal importance sampling distribution, $g(\omega)$, has to be roughly proportional to $c(\omega)f(\omega)$. Seeing how $g(\omega)$ is known up to constant, I am wondering whether I can use MCMC along with the proposal distribution $c(\omega)f(\omega)$ to eventually generate samples from $g(\omega)$ .
My questions here are:
Can MCMC be used within this setting? If so, what MCMC method would be appropriate? I am working in MATLAB, so I have a preference to anything that already has a MATLAB implementation.
Are there any techniques that I can use to speed up the burn-in period for MCMC. And how can I tell that the stationary distribution has been reached? In this case, it actually takes a fair bit of time to calculate $c(\omega)$ for a given $\omega$.