# Does intlinprog find a local minimum or global minimum?

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ryugasen am 8 Nov. 2021
Bearbeitet: Matt J am 8 Nov. 2021
I have a problem that can mathematically be described as a MILP. However, it is essential that I find the global optimum (if it exists) and not just a feasible point. So I understand that the relevant matlab function is intlinprog. According to Matlab documentation (https://www.mathworks.com/help/optim/ug/local-vs-global-optima.html), I know that linprog guarantees global optimum and not just a local one. My question is does intlinprog guarantee the same or not?
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Matt J am 8 Nov. 2021
Bearbeitet: Matt J am 8 Nov. 2021
In ideal math, yes, however real world computers can't do ideal math, so you can be significantly off from the global solution depending on how you set your tolerances.. This is illustrated in the example below. The ideal global minimum is y=1, but intlinprog finds y=0, because it is within the default numerical tolerances.
x=optimvar('x',1,'type','integer','LowerBound',0);
y=optimvar('y',1,'type','integer','LowerBound',0);
prob=optimproblem('Objective',y, 'Constraints', y>=x+1e-4);
sol=solve(prob)
Solving problem using intlinprog. LP: Optimal objective value is 1.000000. Optimal solution found. Intlinprog stopped at the root node because the objective value is within a gap tolerance of the optimal value, options.RelativeGapTolerance = 0.0001 (the default value). The intcon variables are integer within tolerance, options.IntegerTolerance = 1e-05 (the default value).
sol = struct with fields:
x: 0 y: 0
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Matt J am 8 Nov. 2021
Bearbeitet: Matt J am 8 Nov. 2021
I don't think so, because let's apply your proposal to my earlier example, adding an upper bound of 2 on both x and y.
x=optimvar('x',1,'type','integer','LowerBound',0,'UpperBound',2);
y=optimvar('y',1,'type','integer','LowerBound',0,'UpperBound',2);
prob=optimproblem('Objective',y, 'Constraints', y>=x+1e-8);
In this case, there are no non-integer variables, so your method reduces to simply evaluating all 9 combinations 0<=(x,y)<=2.
But how will you decide whether the combination (x,y)=(0,0) is supposed to be feasible or not, bearing in mind that the 1e-8 might just be floating point noise? The decision you make will change the solution and its optimum value by 1.

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Chunru am 8 Nov. 2021
MILP is NP-hard problem. All solvers require some heuristic rules and the global optimum can not guarenteed for larger problems.
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John D'Errico am 8 Nov. 2021
Another issue is that it is easy to formulate a problem with multiple solutions, all equally good. intlinprog should find one of them, but any solution is as good as another. These will typically lie along a constraint boundary, or parallel to one. So is that a global solution or a local one? It depends on how you choose to define what a local solution means to you.
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