# 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:** 6

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**Author:** ![jeffreyesun](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/jeffreyesun/32/13379_2.png) [@jeffreyesun](https://discourse.julialang.org/u/jeffreyesun)\
**Post date:** [May 8, 2020, 6:27pm UTC](https://discourse.julialang.org/t/a-hacky-guide-to-using-automatic-differentiation-in-nested-optimization-problems/39123/6 "2020-05-08T18:27:17Z")

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You’re absolutely right that nested optimization is not the best solution for the example I gave. However, there are problems for which nesting optimizations is probably best. I intended this more as a general method for dealing with nested uses of `optimize` when you want to use forward autodiff. In my research, for example (and the impetus for trying to get this to work), I’m trying to solve

\begin{aligned} &\min\_x \left(\sum\_{i=1}^n f\_i(x)^\sigma\right)^\frac{1}{\sigma}\\ \text{where}\quad f\_i(x) &= \min\_zh(x,z,d\_i) \end{aligned}

where the x, d, and z are all vectors, and n\approx 100. In this case, solving for all the z in a single `optimize` is infeasible–I would have an input vector several hundred entries long.

This sort of thing comes up a lot in economics (esp. macro/IO/finance), where you have a low-dimensional set of parameters \theta and a large set of decision-makers i who make decisions z\_i based on \theta.

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