# Simulation optimisation: Monte carlo simulation, regression, optimise within regression model?

Can you help me identify if this technique has a standard name to help me explore the literature?

Suppose I have a black-box stochastic simulation parameterised by $$X=[x_1,...,x_p]$$ with some single output measure $$y$$ and wish to minimize $$E[y]$$ within some bounded parameter region $$\Theta \subset \mathbb{R}^p$$. However the simulator is computationally expensive and a limited number of runs are possible.

Is this a possible method?

1) Produce some Latin hypercube of $$N$$ parameter points, $$\theta_1,...,\theta_n$$

2) Perform simulations for each parameter point $$\theta_i$$, each with $$R$$ replications, resulting in $$y_{ir}$$

3) Estimate a regression model $$y \sim \theta$$

(Assumption: making "predictions" with this regression model is computationally efficient compared to running the original simulator)

4) Apply standard optimisation search methods to find $$\theta^* \in \Theta$$ by "predicting" $$\theta$$ using regression model (instead of simulator)

So the regression model (hopefully) "approximates" the simulation and so optimising within the regression model (hopefully) optimises approximately within the simulation.

Is this methods a) a good idea; b) a well recognized method; and if so c) what name does it go by in literature?

Does "Kriging" replace the regression model? (Is the regression model a "meta-model"?)

EDIT: So "Surrogate Based Model Optimisation" (SBMO) is the field I was looking for.

Follow on question: how applicable are the methods for stochastic simulations? i.e. for fixed parameters $$\theta$$ I get different results out of the simulator. The sample mean can be taken as a point estimate of the expectation but a large number of samples might be necessary to make it stable.

## 2 Answers

After further reading it seems that "surrogate models" do exactly this (aka. "meta-models"), a recent discussion of methods is here:

https://onlinelibrary.wiley.com/doi/full/10.1111/itor.12292

Also called "Surrogate model-based optimisation" (SMBO)

Survey of methods: https://martinzaefferer.de/wp-content/uploads/2016/10/survey-1.pdf

You might want to google "response surface methodology" and "simulation" together. There is a ton of literature on this, a mixture of how do to response surface modeling (and how to use it to optimize system parameters) and examples of its application.