# Optimization convergence requirements

**URL:** <https://discourse.julialang.org/t/optimization-convergence-requirements/139122>\
**Category:** Specific Domains\
**Tags:** optim\
**Created:** [August 31, 2026, 7:06pm UTC](https://discourse.julialang.org/t/optimization-convergence-requirements/139122 "2026-08-31T19:06:48Z")\
**Posts on this page:** 6\
**Page:** 1

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**Author:** ![dgagnon](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/dgagnon/32/45466_2.png) [@dgagnon](https://discourse.julialang.org/u/dgagnon)\
**Post date:** [August 31, 2026, 7:06pm UTC](https://discourse.julialang.org/t/optimization-convergence-requirements/139122/1 "2026-08-31T19:06:48Z")

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Hello

I have a function from [0,1] to [0,1], which I can shown to always have a unique maximum. I have been using Optim.jl with the default Brent’s method to compute it.  
I am a statistician, not a specialist on numerical algorithms. But I still need to confirm that this algorithm will never fail me. I see a reference to Brent’s book and I guess I will have to read parts of it. However, I would like some pointers to where I can find criteria for garanteed convergence. Further, I can also compute the derivate and can provide it to an algorithm requiring it if need be. I don’t know which algorithm to choose. My function is well behaved, I just need to plug into my article a refrence to Optim.jl, the algorithm I use, and a justification as to why the way I optimise is valid. I don’t care about speed of convergence.  
If any interest, it is for a maximum likelihood estimate. I’d rather spend my time on its asymtotic behaviour then on numerical calculation aspects, I’m more familiar with the former than the later.

Thanks for your help.

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**Author:** ![PeterSimon](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/petersimon/32/25193_2.png) [@PeterSimon](https://discourse.julialang.org/u/PeterSimon)\
**Post date:** [August 31, 2026, 7:15pm UTC](https://discourse.julialang.org/t/optimization-convergence-requirements/139122/2 "2026-08-31T19:15:00Z")

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Is the function convex? If not, is there only a single relative maximum within the initial bracketing interval? As I understand it, if either of these is true, then convergence is guaranteed with Brent’s method.

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**Author:** ![dgagnon](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/dgagnon/32/45466_2.png) [@dgagnon](https://discourse.julialang.org/u/dgagnon)\
**Post date:** [August 31, 2026, 7:42pm UTC](https://discourse.julialang.org/t/optimization-convergence-requirements/139122/3 "2026-08-31T19:42:35Z")

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Thanks a lot  
I’m pretty sure the log of the likelihood is convex. I’ll check. But I already know that there are no local maximum. Second derivative is always negative.  
Could you point my to a reference for this? I will need to plug it in my text with a reference so I don’t get any challenge on this.

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**Author:** ![PeterSimon](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/petersimon/32/25193_2.png) [@PeterSimon](https://discourse.julialang.org/u/PeterSimon)\
**Post date:** [August 31, 2026, 9:36pm UTC](https://discourse.julialang.org/t/optimization-convergence-requirements/139122/4 "2026-08-31T21:36:54Z")

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A good reference is the book: Brent, Richard P. _Algorithms for minimization without derivatives_. Dover, 2002. This is a reissue of the original 1973 Prentiss-Hall book.

Brent discusses the convergence of the method in Section 5.5 of the book. He requires the function to be unimodal to guarantee convergence to the global extremum.

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**Author:** ![PeterSimon](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/petersimon/32/25193_2.png) [@PeterSimon](https://discourse.julialang.org/u/PeterSimon)\
**Post date:** [August 31, 2026, 9:43pm UTC](https://discourse.julialang.org/t/optimization-convergence-requirements/139122/5 "2026-08-31T21:43:03Z")

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P.S. The Kindle edition of the book can be purchased from Amazon for about $8 US.

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**Author:** ![dgagnon](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/dgagnon/32/45466_2.png) [@dgagnon](https://discourse.julialang.org/u/dgagnon)\
**Post date:** [September 1, 2026, 10:23am UTC](https://discourse.julialang.org/t/optimization-convergence-requirements/139122/6 "2026-09-01T10:23:47Z")

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Thanks so much.

I’m an old man, I get the paperback version and add it to my other Dover books.