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.