# integration_design_optim: Function to build integration points (for the SUR criterion) In GPareto: Gaussian Processes for Pareto Front Estimation and Optimization

## Description

Modification of the function `integration_design` from the package `KrigInv-package` to be usable for SUR-based optimization. Handles two or three objectives. Available important sampling schemes: none so far.

## Usage

 ```1 2 3 4 5 6 7 8``` ```integration_design_optim( SURcontrol = NULL, d = NULL, lower, upper, model = NULL, min.prob = 0.001 ) ```

## Arguments

 `SURcontrol` Optional list specifying the procedure to build the integration points and weights. Many options are possible; see 'Details'. `d` The dimension of the input set. If not provided `d` is set equal to the length of `lower`. `lower` Vector containing the lower bounds of the design space. `upper` Vector containing the upper bounds of the design space. `model` A list of kriging models of `km` class. `min.prob` This argument applies only when importance sampling distributions are chosen. For numerical reasons we give a minimum probability for a point to belong to the importance sample. This avoids probabilities equal to zero and importance sampling weights equal to infinity. In an importance sample of `M` points, the maximum weight becomes `1/min.prob * 1/M`.

## Details

The `SURcontrol` argument is a list with possible entries `integration.points`, `integration.weights`, `n.points`, `n.candidates`, `distrib`, `init.distrib` and `init.distrib.spec`. It can be used in one of the three following ways:

• A) If nothing is specified, `100 * d` points are chosen using the Sobol sequence;

• B) One can directly set the field `integration.points` (`p * d` matrix) for prespecified integration points. In this case these integration points and the corresponding vector `integration.weights` will be used for all the iterations of the algorithm;

• C) If the field `integration.points` is not set then the integration points are renewed at each iteration. In that case one can control the number of integration points `n.points` (default: `100*d`) and a specific distribution `distrib`. Possible values for distrib are: "`sobol`", "`MC`" and "`SUR`" (default: "`sobol`"):

• C.1) The choice "`sobol`" corresponds to integration points chosen with the Sobol sequence in dimension `d` (uniform weight);

• C.2) The choice "`MC`" corresponds to points chosen randomly, uniformly on the domain;

• C.3) The choice "`SUR`" corresponds to importance sampling distributions (unequal weights).
When important sampling procedures are chosen, `n.points` points are chosen using importance sampling among a discrete set of `n.candidates` points (default: `n.points*10`) which are distributed according to a distribution `init.distrib` (default: "`sobol`"). Possible values for `init.distrib` are the space filling distributions "`sobol`" and "`MC`" or an user defined distribution "`spec`". The "`sobol`" and "`MC`" choices correspond to quasi random and random points in the domain. If the "`spec`" value is chosen the user must fill in manually the field `init.distrib.spec` to specify himself a `n.candidates * d` matrix of points in dimension `d`.

## Value

A list with components:

• `integration.points` `p x d` matrix of p points used for the numerical calculation of integrals

• `integration.weights` a vector of size `p` corresponding to the weight of each point. If all the points are equally weighted, `integration.weights` is set to `NULL`

## References

V. Picheny (2014), Multiobjective optimization using Gaussian process emulators via stepwise uncertainty reduction, Statistics and Computing.

`GParetoptim` `crit_SUR` `integration_design`