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

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.

[David Hill]

After reviewing the documentation, https://www.mathworks.com/help/optim/ug/fmincon.html, if you have a specific question, show us your code and ask.

[Bill Masters]

Hello,

I found fseminf useful to solve my problem, but there is another issue with this toolbox:

the additional variable must be vectors of at most length two! where in my problem, design variable x and the additional variable y are of the same dimension that could be a reletively large number. I'm confused which toolbox to use and how!

objfun= @(x) C'*x+rho*(norm(AA*x-z-pp+u))^2;  %objective function (variable: x)
constaint1= @(x,y) -B*x+subplus(B)*pp+M*(y-ones(L*D,1));    %constraint of two variable x and y with the same dimensions
constaint2= @(x,y)  B*x+subplus(-1*B)*pp+M*y;
constraint3= @(y)   y-y^2
lb1=zeros(n,1)   %lower bound of variable x
lb2=zeros(n,1)   %lower bound of variable y
ub=ones(n,1)     %upper bound of variable y

[Bill Masters]

does anybody know whether this problem could be solved using CVX toolbox or not?!

Answer 1

[John D'Errico]

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]

Hi, thank you for your reply.

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