# Preconditioner for nonlinear optimization

**URL:** <https://discourse.julialang.org/t/preconditioner-for-nonlinear-optimization/43558>\
**Category:** Optimization (Mathematical)\
**Created:** [July 23, 2020, 3:14pm UTC](https://discourse.julialang.org/t/preconditioner-for-nonlinear-optimization/43558 "2020-07-23T15:14:13Z")\
**Posts on this page:** 1\
**Page:** 1

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**Author:** ![mleprovost](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/mleprovost/32/7166_2.png) [@mleprovost](https://discourse.julialang.org/u/mleprovost)\
**Post date:** [July 23, 2020, 3:14pm UTC](https://discourse.julialang.org/t/preconditioner-for-nonlinear-optimization/43558/1 "2020-07-23T15:14:13Z")

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Hello,

I am working on the following nonlinear minimization problem:

\displaystyle{\min\_{\boldsymbol{c} \in R^n}} \mbox{ }\frac{1}{M} \sum\_{i=1}^{M} \left(\frac{1}{2} \left[\boldsymbol{c}^T \boldsymbol{f}\_0 + \int\_0^{x^i} \sigma(\boldsymbol{c}^T\boldsymbol{g}(t)) dt \right]^2 - \log(\sigma(\boldsymbol{c}^T \boldsymbol{g}(x^i)) \right), where \sigma(x) = \log(1 + \exp(x)) is the softplus function. x^1,x^2,\ldots, x^M are parameters with M \sim 200-300 and the coefficient \boldsymbol{c} has dimension \sim 20. This optimization is a critical part of my code.

For now, I am using Optim.jl with L-BFGS.  
Do you have some advice to speed-up the optimization?  
Are there some rules to design a pre-conditioner?  
Should I supply the Hessian as well and move to a Newton method?

Thank you for your help,