# A hacky guide to using automatic differentiation in nested optimization problems

**URL:** <https://discourse.julialang.org/t/a-hacky-guide-to-using-automatic-differentiation-in-nested-optimization-problems/39123>\
**Category:** Optimization (Mathematical)\
**Tags:** economics, forwarddiff\
**Created:** [May 8, 2020, 5:29pm UTC](https://discourse.julialang.org/t/a-hacky-guide-to-using-automatic-differentiation-in-nested-optimization-problems/39123 "2020-05-08T17:29:59Z")\
**Posts on this page:** 1\
**Showing post:** 5

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**Author:** ![stevengj](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/stevengj/32/71_2.png) [@stevengj](https://discourse.julialang.org/u/stevengj)\
**Post date:** [May 8, 2020, 6:07pm UTC](https://discourse.julialang.org/t/a-hacky-guide-to-using-automatic-differentiation-in-nested-optimization-problems/39123/5 "2020-05-08T18:07:11Z")

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First of all, if you are solving \min\_x \left[\min\_z h(x,z) \right], then you should simply combine it into

\min\_{x,z} h(x,z)

i.e. minimize over x and z simultaneously. Assuming h(x,z) is smooth, you can then apply any mullti-variate minimization approach that you want. Adding a constraint g(x,z) = 0, or any other differentiable constraint, is also easy.

In contrast, directly nesting calls to `optimize` as in your suggestion is probably suboptimal. You will waste a lot of time optimizing the “inner” problem to high precision to get f(x), because you immediately throw away this effort by updating x.

A tricker case to handle is the “minimax” problem

\min\_{x\in X} \left[\max\_{z\in Z} h(x,z) \right].

If Z is finite set (or can be discretized/approximated by a finite set), one approach is an “epigraph” formulation, which transforms it to an _exactly_ equivalent problem with differentiable constraints by adding a “dummy” variable t:

\min\_{x\in X, t \in \mathbb{R}} t \\ \mbox{subject to } t \ge h(x,z) \mbox{ for } z \in Z

at which point you can apply various nonlinear programming algorithms. (One of my favorites is the CCSA algorithm, which can be found in NLopt; I keep meaning to put together a pure-Julia version of this algorithm, since it is so simple and flexible.) If Z is a continuous set, however, matters are more complicated — there are a variety of “continuous minimax” algorithms proposed in the literature for this case, but libraries are harder to find.

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