---
title: "Dealing with constraints"
author: "Fabrice Zaoui"
date: "October 05 2018"
output: html_document
vignette: >
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteIndexEntry{A constrained problem}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```
# Short Description

**caRamel** is a multiobjective evolutionary algorithm combining the MEAS algorithm and the NGSA-II algorithm.

Download the package from CRAN or [GitHub](https://github.com/fzao/caRamel) and then install and load it.

Dealing with constraints is possible with **caRamel** by returning a NaN value for an infeasible solution. See the example below.

```{r caRa}
library(caRamel)
```

# Test functions

## Constr-Ex problem

[*Constr-Ex*](https://en.wikipedia.org/wiki/File:Constr-Ex_problem.pdf) test function has two objectives with two variables and two inequality constraints.

```{r constr}
constr_ex <- function(i) {
  # functions f1 and f2
  s1 <- x[i,1]
  s2 <- (1. + x[i,2]) / x[i,1]
  # now test for the feasibility
  # constraint g1
  if((x[i,2] + 9. * x[i,1] - 6.) < 0. | (-x[i,2] + 9. * x[i,1] -1.) < 0.) {
    s1 <- NaN
    s2 <- NaN
  }
  return(c(s1, s2))
}
```

Note that :

* parameter _i_ is mandatory for the management of parallelism.
* the variable __must be named__ _x_ and is a matrix of size [npopulation, nvariables].

The variable lies in the range [0.1, 1] and [0, 5]:

```{r constr_variable}
nvar <- 2 # number of variables
bounds <- matrix(data = 0., nrow = nvar, ncol = 2) # upper and lower bounds
bounds[1, 1] <- 0.1
bounds[1, 2] <- 1.
bounds[2, 1] <- 0.
bounds[2, 2] <- 5.
```

Both functions are to be minimized:

```{r constr_objectives}
nobj <- 2 # number of objectives
minmax <- c(FALSE, FALSE) # min and min
```

Before calling **caRamel** in order to optimize the Constr_Ex problem, some algorithmic parameters need to be set:

```{r constr_param}
popsize <- 100 # size of the genetic population
archsize <- 100 # size of the archive for the Pareto front
maxrun <- 1000 # maximum number of calls
prec <- matrix(1.e-3, nrow = 1, ncol = nobj) # accuracy for the convergence phase
```

Then the minimization problem can be launched:

```{r schaffer_launch, fig.show="hide", results="hide"}
results <-
  caRamel(nobj,
          nvar,
          minmax,
          bounds,
          constr_ex,
          popsize,
          archsize,
          maxrun,
          prec,
          carallel=FALSE) # no parallelism
```

Test if the convergence is successful:

```{r schaffer_OK}
print(results$success==TRUE)
```

Plot the Pareto front:

```{r schaffer_plot1}
plot(results$objectives[,1], results$objectives[,2], main="Constr_Ex Pareto front", xlab="Objective #1", ylab="Objective #2")
```

```{r schaffer_plot2}
plot(results$parameters, main="Corresponding values for X", xlab="Element of the archive", ylab="X Variable")
```