I'm solving simple single-objective multidimensional global optimization problem using various stochastic algorithms like Monte-Carlo, GA and other evolutionary approaches. The task is formulated as follows:
- search space $X \subseteq R^n$
- objective function $f: R^n \rightarrow R$
- must find $\textbf{x}^*=\{x_1^*, ..., x_n^* \} \in X: f(\textbf{x}^*)=\min_{\textbf{x} \in X} f(\textbf{x})$
However, I also have some preferences of $x_i \in [a;b]$ which are presented in set of normalized values (from $0$ to $1$). This set can be interpolated (here no matter how) to probability distributions. So, for given $x_i$ exists function $g_i(x)$. Figure below shows typical distribution.
This doesn't mean that $x_i$ always possess preferable values and in peaks exists smaller value of objective. However, better values of $f$ tends to be near peaks.
The question is how to use this information about $x_i$ preferable values during optimization process and acheive better results (e.g. rate of convergence)? Any recommendations/books/articles with similar situations will be appreciated. I even don't know in what math area I should look for.
I have tried to use simulated annealing — just added $g_i$ (multiplied by $-1$) to objective function with a high weight at the beginning and then gradually reduce it to zero. With this simple scheme I didn't acheive much better results.