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Is it possible to find the nearest solution to optimal for a mixed-integer problem? For example I would want the simplified problem below:

f = [1;1;1];
intcon = 1:3;

Aeq = [0.99,0.97,0.15];
beq = 0.16;
lb = zeros(3,1);
ub = [1;1;1]; 

x = intlinprog(f,intcon,[],[],Aeq,beq,lb,ub)

to return x=[0;0;1] as this is the nearest integer solution to the objective value of 0.16. Instead currently it returns

Intlinprog stopped because no point satisfies the constraints.

Does not necessarily have to run intlinprog. Would ideally need to also work if beq is low, for example 0.14.

m7913d
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Mary
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    Don't post Ax=b as constraint, but minimize the objective norm(Ax-b) / e.g. least-squares. Depending on your norm, this might be a mixed-integer linear-program or something else (qp, socp, ...). – sascha Jun 23 '17 at 15:32
  • Thanks. That would also be an option. Do you have an example of applying MI to LS? – Mary Jun 23 '17 at 15:55
  • Not for matlab (as i'm not a matlab-user). intlinprog does not seem to be a good candidate here (as the objective, depending on your norm, might be nonlinear which is not supported, but [this example](https://de.mathworks.com/help/optim/ug/mixed-integer-quadratic-programming-portfolio-optimization.html) talks about linearization approaches. MIQP is not as popular as MI and there are not that much solvers (especially open-source), but every commercial solver will do (gurobi, cplex and co.). – sascha Jun 23 '17 at 16:01

1 Answers1

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You can introduce some slack variables to allow some constraint violation when needed as follows:

largeValue = 100; % choose some large value to penalise the constraint violation
f_ = [f; largeValue; largeValue]; % penalise both slack variables
Aeq_ = [Aeq, 1, -1]; % add a positive and negative slack variable to the constraint
beq_ = beq;
lb_ = [lb; 0; 0]; % limit the constraint to a positive number
ub_ = [ub; inf; inf];

x_ = intlinprog(f_,intcon,[],[],Aeq_,beq_,lb_,ub_); % solve the adapted problem
x = x_(1:3) % extract the solution of the original problem

Remarks

  • I added two (positive) slack variables, one for a positive constraint violation and another one for a negative constraint violation.

  • You should penalise the slack variables with a large value, otherwise it is beneficial to violate your constraints more than strictly necessary. A more general approach would be to determine a good penalisation value based on the values in f and Aeq, for example 

    largeValue = 2*max(abs(f))/min(abs(Aeq(Aeq ~= 0)))
m7913d
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