# How to solve an optimization problem over two variables using fmincon?

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Bill Masters on 5 Apr 2021
Commented: Bill Masters on 7 Apr 2021
Hello, I'm trying to solve an optimization problem in which there is a decision variable in its objective function, and some constraints with afformentioned variable and also another decision variable that must be calculated. this problem could be depicted as below:
Minimize Z=f(x)
subject to: g(x,y)<=0
x and y are both decision variables that in particular y is a binary variable that gonna choose an additional constraint to make it continues.
Bill Masters on 7 Apr 2021
does anybody know whether this problem could be solved using CVX toolbox or not?!

John D'Errico on 5 Apr 2021
Edited: John D'Errico on 5 Apr 2021
You CANNOT use fmincon on a problem with binary variables or any form of discrete variables.
However, IF y is indeed binary, then you have only two cases to consider. So solve the problem to minimize f(x), given y == 0, and then repeat, solving it for the minimum over x of f(x), given y == 1.
Take the better of the two results and you are done. There really is little more than that to do here. You still need to choose intelligent starting values for x of course. Note that if there are multiple disjoint regions for x that satisfy the constraints, thus g(x,y)<=0, then you need to search within EACH of them. Fmincon cannot intelligently jump from one such region to another to search among them all.
If you had a more complicated case where you had multiple binary variables, then you would be forced to use a code like GA, which can handle the fully general problem. But that is apparently not the case here.
Bill Masters on 6 Apr 2021
Edited: Bill Masters on 6 Apr 2021
I'm going to use two constraints that may help with this issue of binary variable:
0<=y<=1, y-y^2<=0
I think using these constraints could be helpful to deal with binary variables.
your first suggestion can not be done because every variable exactly is a vector of variables and searching among all of the possiblilities is not efficient.
however, the first obstacle here is to find a way to solve a nonlinear optimization problem with multiple vector variables.
Min Z=f(x)
s.to: g(x)<=0
h(x,y)<=0
y-y^2<=0
x>=0
0<=y<=1