# An elegant way to construct recursive objective?

**URL:** <https://discourse.julialang.org/t/an-elegant-way-to-construct-recursive-objective/2503>\
**Category:** Optimization (Mathematical)\
**Created:** [March 6, 2017, 5:51pm UTC](https://discourse.julialang.org/t/an-elegant-way-to-construct-recursive-objective/2503 "2017-03-06T17:51:07Z")\
**Posts on this page:** 6\
**Page:** 1

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**Author:** ![xqrongm](https://avatars.discourse-cdn.com/v4/letter/x/d9b06d/32.png) [@xqrongm](https://discourse.julialang.org/u/xqrongm)\
**Post date:** [March 6, 2017, 5:51pm UTC](https://discourse.julialang.org/t/an-elegant-way-to-construct-recursive-objective/2503/1 "2017-03-06T17:51:07Z")

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I need to fit a model on a vector `f` of length `n`, where `f[i]` is recursively calculated from `f[i-1]`. There is another vector `d` of length `n` related to `f` in the following way:  
`f[i] = f0 + d[i] d[1] = d0 + k * log(1 + d0 / k) d[i] = d[i-1] + k * log(1 + d[i-1] / k)`  
Here is the JuMP code I have so far for this model fitting.  
`jmp_model = Model(;kwargs_Model...) @variable(jmp_model, f0) @variable(jmp_model, d0 >= 0) @variable(jmp_model, k)`  
Is there an elegant way to construct the objective for this type of model? Thanks!

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**Author:** ![bbrunaud](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bbrunaud/32/1737_2.png) [@bbrunaud](https://discourse.julialang.org/u/bbrunaud)\
**Post date:** [March 7, 2017, 3:18am UTC](https://discourse.julialang.org/t/an-elegant-way-to-construct-recursive-objective/2503/2 "2017-03-07T03:18:53Z")

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Hello.  
You could define constraints to set the relationships. For example

```julia
m = Model()

@variable(m, f0)
@variable(m, d0 >= 0)
@variable(m, k)

@constraint(m, frel[i in I], f[i] == f0 + d[i])
@constraint(m, dinit, d[1] == d0 + k*log(1 + d0 / k))
@constraint(m, drel[i in 2:I[end]], d[i] == d[i-1] + k * log(1 + d[i-1] / k))

```

However, the standard way to fit a model would be to minimize the sum of squares (for example) of your prediction compared to your observations. So `f` should also be a variable, and you should add an objective like

```julia
@objective(m, Min, sum( (f[i] - fobs[i])^2 for i in I))

```

I hope it helps. regards

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**Author:** ![xqrongm](https://avatars.discourse-cdn.com/v4/letter/x/d9b06d/32.png) [@xqrongm](https://discourse.julialang.org/u/xqrongm)\
**Post date:** [March 7, 2017, 9:23pm UTC](https://discourse.julialang.org/t/an-elegant-way-to-construct-recursive-objective/2503/3 "2017-03-07T21:23:32Z")

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Thank you very much for the help!

You said “`f` should also be a variable”. I guess `d` should also be a variable? And I should define them like this?  
`@variable(m, f[I]) @variable(m, d[I])`  
where `I` is an vector containing all the indices for `fobs`

Thanks again!

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**Author:** ![xqrongm](https://avatars.discourse-cdn.com/v4/letter/x/d9b06d/32.png) [@xqrongm](https://discourse.julialang.org/u/xqrongm)\
**Post date:** [March 7, 2017, 9:24pm UTC](https://discourse.julialang.org/t/an-elegant-way-to-construct-recursive-objective/2503/6 "2017-03-07T21:24:27Z")

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In addition, should I use `@objective` or `@NLobjective`? I’m not sure because `k * log(1 + d[i-1] / k` is non-linear, though sum of squares can be considered linear. Thanks!

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**Author:** ![xqrongm](https://avatars.discourse-cdn.com/v4/letter/x/d9b06d/32.png) [@xqrongm](https://discourse.julialang.org/u/xqrongm)\
**Post date:** [March 8, 2017, 2:37am UTC](https://discourse.julialang.org/t/an-elegant-way-to-construct-recursive-objective/2503/7 "2017-03-08T02:37:14Z")

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Please don’t worry about my follow-up questions any more.

I tried this:  
`@constraint(m, dinit, d[1] == d0 + k*log(1 + d0 / k)) @constraint(m, dinit, d[1] == d0 + k*log(1 + d0 / k)) @objective(...)`  
and got an error asking me to use `@NLconstraint` and `@NLobjective`.

I changed to this:  
`@NLconstraint(m, dinit, d[1] == d0 + k*log(1 + d0 / k)) @NLconstraint(m, dinit, d[1] == d0 + k*log(1 + d0 / k)) @NLobjective(...)`  
and it returned successfully.

In addition, I used:  
`@variable(m, f[I]) @variable(m, d[I])`  
where `I` is an vector containing all the indices for `fobs`

And I think I got the model fitting to work. Thanks again!

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<div class="post-metadata">

**Author:** ![bbrunaud](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bbrunaud/32/1737_2.png) [@bbrunaud](https://discourse.julialang.org/u/bbrunaud)\
**Post date:** [March 20, 2017, 4:33am UTC](https://discourse.julialang.org/t/an-elegant-way-to-construct-recursive-objective/2503/8 "2017-03-20T04:33:51Z")

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I’m glad it worked. Personally, I would prefer that `@NLconstraint` and `@NLobjective` wouldn’t exist. It might be confusing and no other AML requires you to explicit that. I imagine there is a good reason why both are required.

regards!