How to deal with discrete constraint fmincon

Hello! I'm writing the code for an optimization problem, and I'm having problem writing the equality constraints.

I have to write these constraints (discrete) and I don't know how to do it. I'm trying to write them into "Aeq", but I don't know how to handle this "sk+1" and "vk+1". Any ideas? Thank you

Answer 1

[Aurele Turnes]

You can write the equality constraint matrix by keeping track of the indices, or you can use the new problem-based workflow to define these constraints more easily.

https://www.mathworks.com/help/optim/ug/problem-based-workflow.html

Start by defining your variables for say N = 10

N = 10;
% Some made-up data
tau = 0.1;
vlower = 0;
vupper = 1;
s0 = 1;
sF = 2;
v0 = 0.1;
vF = 0.8;

% Define variables
a = optimvar('a', N);
s = optimvar('s', N);
v = optimvar('v', N, 'LowerBound', vlower, 'UpperBound', vupper);

Then your constraints are simply:

conS = s(2:end) == s(1:end-1) + tau*v(1:end-1);
conV = v(2:end) == v(1:end-1) + tau*a(1:end-1);
conInitS = s(1) == s0;
conFinalS = s(end) == sF;
conInitV = v(1) == v0;
conFinalV = v(end) == vF;

Note that you could write the same constraints with a for-loop instead, but it is always more efficient to vectorize your code as above.

% conS = optimconstr(N-1,1);
% for i = 1:N-1;
%     conS(i) = s(i+1) == s(i) + tau*v(i);
% end

Then, just create your optimproblem and populate Objective and Constraints.