I am using the R programming language - I am trying to perform "multi objective constrained optimization".

I defined a function ("funct_set") with "4 objectives" ( `f[1], f[2], f[3], f[4]`

) which are to be minimized for a set of "seven inputs" ( `[x1], [x2], [x3], x[4], x[5], x[6], x[7]`

):

```
#load libraries
library(dplyr)
library(mco)
#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 )))
f[1] = -mean(table_a$quant)
f[2] = -mean(table_b$quant)
f[3] = -mean(table_c$quant)
#group all tables
final_table = rbind(table_a, table_b, table_c)
# calculate the total mean : this is what needs to be optimized
f[4] = -mean(final_table$quant)
return (f);
}
```

Next, I define a series of 4 "restrictions" (i.e. logical conditions/constrains) used in the optimization:

```
#define restrictions
restrictions <- function (x) {
x1 <- x[1]; x2 <- x[2]; x3 <- x[3]; x4 <- x[4]; x5<- x[5] ; x6 <- x[6]; x7 <- x[7]
restrictions <- logical(4)
restrictions[1] <- (x3 - x1 >= 0)
restrictions[2] <- (x4 - x2 >= 0)
restrictions[3] <- (x7 - x6 >= 0)
restrictions[4] <- (x6 - x5 >= 0)
return (restrictions);
}
```

Finally, I run the optimization algorithm that attempts to simultaneously minimize all 4 objectives with respect to the restrictions:

```
#run optimization
optimization <- nsga2(funct_set, idim = 7, odim = 4 , constraints = restrictions, 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 works fine.

**Problem** : I noticed that in the output of this code, the optimization algorithm is not respecting the restrictions. For example:

In the above picture, I have identified some rows where the logical conditions specified in the restrictions are violated.

Does anyone know why this is happening? Have I incorrectly specified the restrictions? Can someone please show me how to fix this?

Thanks