I am working with R and trying to perform optimization on the following function using the "mco" library (https://cran.r-project.org/web/packages/mco/mco.pdf):

```
#load libraries
library(mco)
library(dplyr)
# create some data for this example
a1 = rnorm(1000,100,10)
b1 = rnorm(1000,100,5)
c1 = sample.int(1000, 1000, replace = TRUE)
train_data = data.frame(a1,b1,c1)
```

Now, I define the function "funct_set" (7 inputs, 4 outputs) for optimization - in this function, I want to set "constraints" such that " `n5, n6, n7`

" can not be smaller than 200. Do this is, I tried to "trick" the computer into assigning the "outputs" (i.e. `f[1], f[2], f[3], f[4]`

) some illogical value (e.g. 99999):

```
#define function
funct_set <- function (x) {
x1 <- x[1]; x2 <- x[2]; x3 <- x[3] ; x4 <- x[4]; x5 <- x[5]; x6 <- x[6]; x[7] <- x[7]
f <- numeric(4)
#bin data according to random criteria
train_data <- train_data %>%
mutate(cat = ifelse(a1 <= x1 & b1 <= x3, "a",
ifelse(a1 <= x2 & b1 <= x4, "b", "c")))
train_data$cat = as.factor(train_data$cat)
#new splits
a_table = train_data %>%
filter(cat == "a") %>%
select(a1, b1, c1, cat)
b_table = train_data %>%
filter(cat == "b") %>%
select(a1, b1, c1, cat)
c_table = train_data %>%
filter(cat == "c") %>%
select(a1, b1, c1, cat)
#calculate quantile ("quant") for each bin
table_a = data.frame(a_table%>% group_by(cat) %>%
mutate(quant = ifelse(c1 > x[5],1,0 )))
table_b = data.frame(b_table%>% group_by(cat) %>%
mutate(quant = ifelse(c1 > x[6],1,0 )))
table_c = data.frame(c_table%>% group_by(cat) %>%
mutate(quant = ifelse(c1 > x[7],1,0 )))
#group all tables
final_table = rbind(table_a, table_b, table_c)
# calculate the total mean
#count number of rows in each table
n5 = data.frame(table_a %>%
summarise(count = n()))
n6 = data.frame(table_b %>%
summarise(count = n()))
n7 = data.frame(table_c %>%
summarise(count = n()))
if (n5 <200){
f[1]=-999999}
else {
f[1] = mean(table_a$quant)
}
if (n6 <200){
f[2]=-999999}
else {
f[2] = mean(table_b$quant)
}
if (n7 <200){
f[3]=-999999}
else {
f[3] = mean(table_c$quant)
}
#f[2] = mean(table_b$quant)
#f[3] = mean(table_c$quant)
f[4] = mean(final_table$quant)
return (f);
}
```

The "mco" library offers a standard way to add constraints to the inputs of the function you are optimizing (i.e. `x[1], x[2], x[3], x[4], x[5], x[6], x[7]`

):

```
#add constraints
gn <- function(x) {
g1 <- x[3] - x[1]
g2<- x[4] - x[2]
g3 <- x[7] - x[6]
g4 <- x[6] - x[5]
#g5 <- n5 > 200
#g6 <- n6 > 200
#g7 <- n7 > 200
return(c(g1,g2,g3,g4))
}
```

But since " `n5, n6, n7`

" are not the final outputs of the function ("funct_set"), constraints can not be placed on them within the " `gn`

" object. This why I tried to define these constraints within the original function ("funct_set") itself.

Finally, I ran the optimization:

```
#run optimization:
optimization <- nsga2(funct_set, idim = 7, odim = 4 , constraints = gn, cdim = 4,
generations=150,
popsize=100,
cprob=0.7,
cdist=20,
mprob=0.2,
mdist=20,
lower.bounds=rep(80,80,80,80, 100,200,300),
upper.bounds=rep(120,120,120,120,200,300,400)
)
```

The above code seems to run successfully, as well as having respected the logical constraints:

```
#view results
#optimized input parameters
head(optimization$par)
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 96.33968 102.80793 103.3724 103.8658 116.9360 119.4670 119.9997
[2,] 102.42030 100.27308 104.6474 105.9168 119.5517 119.9530 119.9992
[3,] 92.77710 100.52490 105.4731 100.6363 108.3434 119.6574 119.9990
#view optimized outputs
head(optimization$value)
[,1] [,2] [,3] [,4]
[1,] 0.8982456 8.038278e-01 0.8833992 0.871
[2,] 0.8546169 9.999990e+05 0.8820961 0.870
[3,] 0.9061033 9.999990e+05 0.8522167 0.871
```

**My Question:** Can someone please tell me if the way I have placed constraints on " * *n5, n6, n7** " is correct? Are there any other ways that allow for these constraints to be placed?

Thanks