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Hello,

I am trying to solve a nonlinear optimization problem using fmincon interior point method. Originally my problem formulation does not have bounds on the decision variable, and when i try to run it without the bounds then it takes infintie time and when I run it with bounds then it is much faster. Following are my questions:

1) Why bounds are making the algorithm faster?

2) The final optimal result for the problem is nowhere near the bound, but my lagrange multiplier for the bounds is coming to be non zero, arent they supposed to be zero if the solution is not hitting the bounds?

3) How is the first order optimality criteria defined for interior point method? I saw the documentation but it is not clear to me, is the infinite norm of the grad or some other equation?

I am giving very good initial guess (exact true values) to my problem to make sure it is near the optimal. When I do that, the optimizer is just giving me the intial guess as my final solution which is not possible as I am feeding noisy data to my problem.

Thanks in advance

Walter Roberson
on 16 Jan 2021

Generally speaking, functions without bounds can take indefinite time to minimize if the function has an asymptope

| |

___/ --v-+

where the v marks the minimum. But if the function happens to land on the shoulder to the left then the local gradient slopes away from the center and the minimizer can take indefinite time exploring that left slope.

fmincon is a local optimizer: there is no way for it to know that it should spend time climbing the hill to the center to get to a better opportunity. And not every such hill happens to lead to a global minimum. Furthermore, the global minimum can be an indefinitely small part of the graph -- imagine putting something heavy on a section of stiff rubber that has a very elastic center, then unless you were quite close to the indentation you would get no information that it existed.

Walter Roberson
on 16 Jan 2021

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