How to define a binary variable as part of constraints in the optimization problem(fmincon)?

I'm working on an optimization problem that optimizes the cost for a microgrid including Battery Storage Units. I used the fmincon function and wrote all constraints except the nonlinear inequalities c(x) which I indicated below. u_ch(t) is a binary variable and can be "1" when the battery is charging or be "0" when it is discharging. I need to know at which time the battery is charging or discharging so I need to calculate "u_ch(t)".

The problem is that I couldn't manage to define a binary variable in Matlab and use it for fmincon. P_Batt_charge(t) and P_Batt_disharge (t) are also decision variables. End of the optimization, I also need to calculate the values of them for each t. Is there any suggestion how can I write these constraints?

P_Batt_charge(t) ≤ 150kW*u_ch (t)

P_Batt_disharge (t)≤180kW*(1-u_ch) (t))

Answer 1

[Alan Weiss]

fmincon does not accept binary constraints, or any integer constraints. The only solvers that accept a nonlinear objective function or constraint function and integer constraints are ga and surrogateopt.

It is often possible to approximate the costs and constraints using linear objectives and constraints. If you can do that, you can use intlinprog as the solver. For an energy-related example, see Optimal Dispatch of Power Generators: Problem-Based.

Alan Weiss

MATLAB mathematical toolbox documentation

[Mehmet Türker TAKCI]

Thanks for your reply. I have searched a little bit more and managed to understand the theory of surrogateopt. But, I have still some problems. Especially to define nonlinear constraint. I have a simple example that is working correctly.

x = sym('x',[N,1]);

intcon=1;

lb = [0,0,0,0];

ub = [1,8,12,15];

[x,fval,exitflag] = surrogateopt(objfn,lb,ub,intcon)

function ff=objfn(x)

Ctotal= 4*x(1)+x(2)-x(3)+x(4)^2;

ff.Fval =Ctotal;

ff.Ineq = x(1)*x(2)-x(3)^2-9;

end

I want to modify this example to my real code and real objfn and constraints. So, I have just made little changes which include another variable to the objfn. I replaced the number 4 with m. Because in my real function has more than one variable and each variable size like 24X1. So when I add another variable to the objfn it doesn't work. When the surrogateopt calls the function objfn, all variables are deleted. I can't use like objfn(x,m)

x = sym('x',[N,1]);

intcon=1;

lb = [0,0,0,0];

ub = [1,8,12,15];

m=4;

[x,fval,exitflag] = surrogateopt(objfn,lb,ub,intcon)

function ff=objfn(x,m)

Ctotal= m*x(1)+x(2)-x(3)+x(4)^2;

ff.Fval =Ctotal;

ff.Ineq = x(1)*x(2)-x(3)^2-9;

end

The code above doesn't work. I also tried a different version of calling objfn but I couldn't achieve it. To sum up, my objective function depends not only X but also on other variables. So, I need to define these variables while calling objfn but the surrogateopt doesn't allow it.

Do you have any suggestions?

[Alan Weiss]

You need to understand how to formulate any problem for Optimization Toolbox or Global Optimization Toolbox. When using the solver-based approach as you are doing, you must put all optimization variables into a single vector, usually called x. See, for example, Writing Scalar Objective Functions. And by the way, you cannot use any symbolic variables, so your sym call is entirely unnecessary.

For example, for your m variable, you must represent that in your x as, say, x(5). So you would have

function ff=objfn(x)
 Ctotal= x(5)*x(1)+x(2)-x(3)+x(4)^2;
 
 ff.Fval = Ctotal;
 ff.Ineq = x(1)*x(2)-x(3)^2-9;
 
end

At the risk of confusing you, if you have MATLAB version R2022b, you can use the problem-based approach. There you can have separate optimization variables and use surrogateopt for solving the problem. See the first Release Note for details. However, for all earlier versions of MATLAB, you must use the solver-based aproach for surrogateopt, and put all your variables into x.

Alan Weiss

MATLAB mathematical toolbox documentation

[Mehmet Türker TAKCI]

Thanks for your reply.Your comments have been very instructive. In my case, m will be the grid price which changes during the hours in a day. For the above example, I just wanted to simulate it with a simple example. So, It will be constant not a decision variable that's why I didn't put it into the x. But, as I understood that even if the constant value which is used for the objective function will be inside the x vector. It is so confusing because I think I need to then revise the matrix A,Aeq or lb or ub. Am I right? Because LB<=x<=UB or Aeq*x=beq.

For the example above,m is equal to 4 and it is represented as x(5), so LB and UB will be:

lb = [0,0,0,0,4];

ub = [1,8,12,15,4];

Are these correct?

I would like to also ask about an error that occurs when I tried to use nonlinear constraints. I used an example as in the link: https://www.mathworks.com/help/gads/convert-between-scalar-and-structure.html

objconstr = packfcn(@objFcn,@nonlincon);

[x,fval] = surrogateopt(objconstr,LB,UB,intcon,A,b,Aeq,beq)

% Error using surrogateopt

% Too many input arguments.

The syntax for the usage surrogateopt is like above. But does it only for linear constraints. Because when I open the function "urrogateopt" , I saw the explanation like:

On the other hand when I changed the "objconstr" as only include the objection function not any nonlinear constraints, I looked into the function's inside and saw that :

In my example, when I delete the ,A,b,Aeq,beq then it says "not enough input parameters". So I didn't understand what is the problem :) Do I need to write my other constraints inside the ff.Ineq ?

[Alan Weiss]

If m is an extra parameter, not a decision variable, include it as an extra parameter as described here.

Alan Weiss

MATLAB mathematical toolbox documentation

[Mehmet Türker TAKCI]

Hi Alan Weiss, many thanks for your help. I will also examine the extra parameter options. But, the other problem is that usage of linear and nonlinear constraints together. I tried to explain my above response but could you please look at this code? (by the way this will be my last question, ı promise:) )

intcon=1;

lb = [0,0,0,0]';

ub = [1,8,12,15]';

[x,fval,exitflag] = surrogateopt(@objconstr,lb,ub,intcon)

function f=objconstr(x)

f.Fval = x(1)+x(2)-x(3)+x(4)^2;

f.Ineq=x(1)*x(2)-x(3)^2-9;

end

The code above is working correctly. x and fval can be found. But, when I add linear contraints like shown below, It gives error.

intcon=1;

lb = [0,0,0,0]';

ub = [1,8,12,15]';

Aeq=[1,1,1,1];

beq=[14];

A=[];

b=[];

[x,fval,exitflag] = surrogateopt(@objconstr,lb,ub,intcon,A,b,Aeq,beq)

function f=objconstr(x)

f.Fval = x(1)+x(2)-x(3)+x(4)^2;

f.Ineq=x(1)*x(2)-x(3)^2-9;

end

So, what is the reason? Is it not possible to use linear and non-linear constraint at the same time?

[Alan Weiss]

As the Release Notes state, linear constraints were introduced to surrogateopt in R2021a. Without seeing the error, I suspect that you have an older version.

Alan Weiss

MATLAB mathematical toolbox documentation

[Mehmet Türker TAKCI]

Hi, yes I use R2020b. So, I need to install R2021a or above to use all types of constraints. Ok, many thanks for all information.