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Greetings,

- MY PROBLEM) I'm using fmincon's interior point algorithm. I want to stop the computation if the distance f(x_k) - f(x_opt) <= epsilon where x_opt is the optimum and x_k is the point generated by the algorithm at iteration k.
- MY TRIALS) I'm seaching for an option to stop computation when the 'mu' of the problemmin f(x) - mu (sum BarrierFunctions(x))is less than a fixed parameter but I don't have found nothing.

I've read the documentation on the stopping criterias, TolFun doesn't seems to do what I want and the only things that refers to the 'mu' is the starting value. Thanks for your help.

Alan Weiss
on 25 Mar 2014

But usually you don't know x_opt. I mean, if you did, then why are you running an optimization? And if you don't know x_opt, then how can you possibly write a stopping criterion that depends on x_opt?

Alan Weiss

MATLAB mathematical toolbox documentation

Matt J
on 25 Mar 2014

Edited: Matt J
on 25 Mar 2014

(1) For each x_k, you will need to find a minorizing surrogate function Q(x;x_k), i.e., it satisfies Q(y;x_k)<=f(y) for all y with equality at y=x_k. You must find such a function that is easy to minimize analytically over the feasible set. Then,

f(x_k) - f(x_opt)<=f(x_k)- min_y Q(y;x_k)

For a good choice of Q(), the right hand side will --->0 as x_k-->x_opt and you can monitor when it falls below your desired epsilon.

For example, in unconstrained problems in which a global lower bound on the Hessian eigenvalues L is known, you could construct the quadratic surrogate

Q(y;x_k)= f(x_k)+ dot(grad(x_k),y-x_k)+ L/2*norm(y-x_k)^2

whose minimum is f(x_k)-grad(x_k)/(2*L). This leads to the upper bound

f(x_k) - f(x_opt) <= grad(x_k)/(2*L)

so you would stop when grad(x_k)<= 2*L*epsilon.

(2) You should be able to reverse engineer mu from the Lagrange Multiplier equations for sub-problem (6-51) in the interior point algorithm documentation. Note that in addition to knowing the minimizing x, you also know the minimizing slacks s_i from the constraint g(x)+s=0.

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