C++ library for nonlinear constrained minimization

Question

I am currently trying to solve nonlinear constrained minimization problem as implemented in matlab "fmincon" function. My expectations are, minimize(fun1,x0,uB,lB,fun2) where x0 is initial state, fun1 is function that needs to be minimized, uB are upper bounds, lB are lower bounds and fun2 is function that provides vectors of nonlinear equalities/inequalities as described in http://www.mathworks.com/help/optim/ug/fmincon.html as nonlcon function. These vectors are changing through iterations as well (they are non-linearly dependent on x_n, n-th iteration of solution vector). In matlab implementation they are in a form c(x)<=0. This is the last piece of code that need to be ported from matlab to c++ and I ve been struggling a lot while trying to find appropriate c++ library containing this algorithm. This is why I am seeking help here and I would much appreciate if you could provide your expertise.

Good example of what I want to do is first one on this page http://www.mathworks.com/help/optim/ug/constrained-nonlinear-optimization-examples.html#f10960?s_tid=doc_12b Only difference is that I need boundaries as well...

Thanks in advance.

Peter

Answer 1

If all you need is a C++ library to solve nonlinear optimization problems, you can use RobOptim. Even though RobOptim was initially developed with robotics optimization problems in mind, it is suitable for any nonlinear optimization problems. It provides a simple C++ interface with plugins for multiple nonlinear solvers (Ipopt, NAG, etc.). Using that kind of wrappers makes it easy to use another NLP solver. If you cannot provide gradients, finite-difference computation can be done automatically.

It's open source so you can check out the source code on GitHub: https://github.com/roboptim/

The analysis done by @Geoff Oxberry is essential for the choice of nonlinear solver that will be called by RobOptim. Note that when dealing with that kind of solvers, parameter tweaking can have a huge impact on performance, and you may still get stuck in local minimas (it really depends on the kind of problem you are dealing with).

Note: I am one of the developers of this project.

Answer 2

If your function is not differentiable, you should be careful about how you use finite differences. If you want to use derivative information, your best bet is probably some sort of semismooth Newton-type method. A set of notes describing such methods can be found here.

Twelve to thirty variables is probably on the upper end of what's doable with pattern search (also called direct search) methods. A recent review paper by Rios and Sahinidis in Journal of Global Optimization on derivative-free optimization methods (such as pattern search methods) can be found here, along with a companion web page. A less recent review paper on these methods by Kolda, Lewis, and Torczon in SIAM Review can be found here. These methods work fairly well with expensive function evaluations, and don't necessarily require differentiability or derivative information.

Many of these methods require some sort of convexity to guarantee convergence to the global optimum, so if you were to solve your problem rigorously, you might need to couple these methods above with a branch-and-bound strategy. However, if you don't care about rigor, an approach like MATLAB's fmincon might work well enough (there are no guarantees anymore). Finite differences will most likely give you a member of the subdifferential of your nondifferentiable function, which could suffice for your problem instance and particular input data to return a sufficiently accurate result for your purposes. In that case, you should probably look at the libraries mentioned in the answers to the question Christian linked in his comment.