Monte Carlo Integration with Computationally expensive Integrant

#1
Consider \(\mathcal{R}_x = \int_{x'} C(x|x')P(x')dx'\), where
\(\mathcal{R}_x\) = risk of following input x
\(C(x|x')\) = cost of following input x when in fact input x' occured
\(P(x')\) = known pdf of inputs evaluated for input x'

Objective : obtain a decent estimate of the integral
Challenge : \(C(x|x')\) is computationally expensive and can only be evaluated for a handful (less than 50) of x,x's.
Context : x is the input of a complex optimization program. To evaluate \(C(x|x')\) we have to find the optimal solution of the optimization program once for x and once for x'. optimal solutions are compared against each other to find their relative costs ,i.e., \(C(x|x')\).