using Bonmin Counne and Ipopt for NLP

Question

I want to just be sure that I am eligible to use Bonmin and Couenne for solving just the NLP problem (Still I do not have integer variable) and I am eager to obtain global optimum not local. I also read that Ipopt first search for the global answer and if it does not find that it will provide a local answer. How I can understand my answer is a global answer when I using Ipopt. Also, I want to what is the best NLP and MINLP open source pythonic solvers for these issues that can be merged with Pyomo? The main reason for my question is the following output using Bonmin:

NOTE: You are using Ipopt by default with the MUMPS linear solver. Other linear solvers might be more efficient (see Ipopt documentation).

Regards

Answer 1

Some notes:

(1) "Ipopt first search for the global answer and if it does not find that it will provide a local answer"
This is probably not how I would phrase it. IPOPT finds local solutions. For some problems these will be the global solution. For convex problems, this is always the case (except for numerical issues).

(2) Bonmin is a local MINLP solver, Couenne is a global NLP/MINLP solver. Typically Bonmin can solve larger problems than Couenne, but you get local solutions.

(3) "NOTE: You are using Ipopt by default with the MUMPS linear solver. Other linear solvers might be more efficient (see Ipopt documentation)."
This is just a notification that you are using IPOPT with linear algebra routines from MUMPS. There are other linear sub-solvers that IPOPT can use and that may perform better on large problems. Often the HARWELL routines (typically called MAnn) give better performance. MUMPS is free while the Harwell routines require a license.

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In a follow-up answer (well it is not answer at all) it is stated:

Regarding Ipopt how I can understand that it is finding the global solution or local optimum? the code will notify that? Regarding to Bonmin according to AMPL page AMPL It provides the global solution for the convex problem " Finds globally optimal solutions to convex nonlinear problems in continuous and discrete variables, and may be applied heuristically to nonconvex problems." And you were saying that it is obtained the local solution, I am a bit confused on this part. But the general question about all those codes is that how I can find out that the answer is global optimum?

(a) Ipopt does not know if a solution is a local or a global optimal solution. For convex problems a local optimum is a global optimal solution. You will need to convince yourself the problem you pass on to Ipopt is convex (Ipopt will not do this for you).

(b) Bonmin: the same: if the problem is convex it will find global solutions. Otherwise you will get a local solution. You will get no notification whether a solution is a global solution: Bonmin does not know if a solution is a global optimum.

(c) When looking for guaranteed global solutions you can use a local solver only when the problem is convex. For other problems you need a global solver. Another approach is to use a multi-start algorithm with a local solver. That gives you confidence that you are not ending up with a bad local optimum.

If possible, I suggest to discuss this with your teacher. These concepts are important to understand (and most solver manuals assume you know about them).