Constraint with decision variable

How can I write a constraint for optimization which includes the decision variable (x)? The constraint is:

max (0, x(i-1) - P) <= x(i) <= min (C, x(i-1) + D)

[John D'Errico]

Don't all constraints for an optimization somehow involve the decision variables? Note that what you have written is actually more than one constraint. There are TWO inequalities there.

Answer 1

[Walter Roberson]

You should use a couple of different kinds of constraints. But you are only constraining variable #i, so we have to initialize the constraints so the other variables are not affected

Nx = length(x);

%no equality constraints
Aeq = [];
beq = [];

%the constraint has to be for some _specific_ i
i = 13;

%now to construct x(i-1) - P <= x(i) and x(i) <= x(i-1) + D
A = zeros(2, Nx);
b = zeros(2, 1);
A(1, i-1) = 1; A(1, i) = -1; b(1) = P;   % (1)  * x(i-1) + (-1) * x(i) <= (P)
A(2, i-1) = -1; A(2, i) = 1; b(2) = D;   % (-1) * x(i-1) + (1)  * x(i) <= (D)

%lower bounds are -inf, upper bounds are +inf for all variables except #i
lb = -inf * ones(1, Nx);
ub = inf * ones(1, Nx);

%but for variable #i, the value cannot be less than 0 or more than C
lb(i) = 0;
ub(i) = C;

fmincon(@objective, x0, A, b, Aeq, beq, lb, ub)

Your constraints cannot be generalized to all of the variables because there is no x(0) to constrain x(1) against.

[Nana]

Hi Walter, thanks for your answer!

So basically, I need two columns of b?

[Walter Roberson]

You need one entry in b for each of the separate constraints. You have two inequality constraints for each i, so two entries in b for each i. It does not matter whether you specify b as a row or as a column vector; a row vector will be transformed into a column vector internally.