What are some of the well-written good references that discusses why finding the penalty parameter when solving a nonlinear constrained optimization problem is hard to find from the computational perspective. What are some of the computational methods done to find the parameter as I understand finding such a parameter is problem-dependant.
Any insights also would be very helpful.
Thanks
Some penalty methods are exact, in the sense that a finite penalty is enough to identify an optimum. Typically the penalty function then needs to be non-differentiable, which might be a bummer. With differentiable penalty functions, such as the augmented Lagrangian, you typically need the penalty to grow indefinitely, but there are those that can be exact without a growing penalty parameter. It however has a cost, in the sense that some constraint qualification holds - which is hard to check.
Here is a paper about just that:
Exact augmented Lagrangian functions for nonlinear semidefinite programming, by Ellen H. Fukuda & Bruno F. Lourenço
Computational Optimization and Applications volume 71, pages 457–482 (2018)
Hope it helps! :-)
I'd suggest IMA Journal of Numerical Analysis(2005) doi: 10.1093/imanum/dri017
The authors give a good technical overview of various approaches.
In actual research, I use FLETCHER, R. & LEYFFER, S.(2002) Nonlinear programming without a penalty function,Mathematical Programming, Series A,91, 239–269.
Their methods work very well for a large class of problems. (I'm maybe biased, though, as Roger and Sven were colleagues of mine in the mid 1990s)
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