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‎README.md

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[![appveyor badge][appveyor_badge]][appveyor_url]
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[![codecov badge][codecov_badge]][codecov_url]
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## Documentation [here][documenter_latest]
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[travis_badge]: https://travis-ci.org/JuliaIntervals/IntervalOptimisation.jl.svg?branch=master
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[travis_url]: https://travis-ci.org/JuliaIntervals/IntervalOptimisation.jl
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[codecov_url]: http://codecov.io/github/JuliaIntervals/IntervalOptimisation.jl?branch=master
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[documenter_stable]: https://JuliaIntervals.github.io/IntervalOptimisation.jl/stable
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[documenter_latest]: https://JuliaIntervals.github.io/IntervalOptimisation.jl/latest
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## Rigorous global optimisation using Julia
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Currently, the package uses an implementation of the Moore-Skelboe algorithm.
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## Usage
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Functions `minimise` and `maximise` are provided to find the **global** minimum or maximum, respectively, of a standard Julia function `f` of one or several variables.
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They return an `Interval` that is guaranteed to contain the global minimum (maximum), and a `Vector` of `Interval`s or `IntervalBox`es whose union contains all the minimisers.
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### Examples
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#### 1D
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```
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using IntervalArithmetic, IntervalOptimisation
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julia> @time global_min, minimisers = minimise(x -> (x^2 - 2)^2, -10..11);
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0.046620 seconds (36.07 k allocations: 1.586 MiB)
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julia> global_min
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[0, 1.50881e-09]
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julia> minimisers
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2-element Array{IntervalArithmetic.Interval{Float64},1}:
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[1.41387, 1.41453]
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[-1.41428, -1.41363]
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```
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#### 2D
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```
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julia> @time global_min, minimisers = minimise( X -> ( (x,y) = X; x^2 + y^2 ),
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(-10000..10001) × (-10000..10001) );
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0.051122 seconds (46.80 k allocations: 2.027 MiB)
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julia> global_min
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[0, 2.33167e-08]
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julia> minimisers
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3-element Array{IntervalArithmetic.IntervalBox{2,Float64},1}:
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[-0.000107974, 0.000488103] × [-0.000107974, 0.000488103]
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[-0.000107974, 0.000488103] × [-0.000704051, -0.000107973]
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[-0.000704051, -0.000107973] × [-0.000107974, 0.000488103]
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```
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Note that the last two `IntervalBox`es do not actually contain the global minimum;
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decreasing the tolerance (maximum allowed box diameter) removes them:
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```
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julia> @time global_min, minimisers = minimise( X -> ( (x,y) = X; x^2 + y^2 ),
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(-10000..10001) × (-10000..10001), 1e-5 );
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0.047196 seconds (50.72 k allocations: 2.180 MiB)
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## Documentation
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Documentation of this package is available [here](https://JuliaIntervals.github.io/IntervalOptimisation.jl/latest/)
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julia> minimisers
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1-element Array{IntervalArithmetic.IntervalBox{2,Float64},1}:
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[-5.52321e-06, 3.79049e-06] × [-5.52321e-06, 3.79049e-06]
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```
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## Author
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‎docs/src/index.md

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# IntervalOptimisation.jl
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# `IntervalOptimisation.jl`
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```@index
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## Rigorous global optimisation using Julia
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This package provides rigorous global optimisation routines written in pure Julia, using interval arithmetic provided by the author's [IntervalArithmetic.jl](https://github.com/JuliaIntervals/IntervalArithmetic.jl) package.
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Currently, the package uses an implementation of the Moore-Skelboe algorithm.
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## Usage
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Functions `minimise` and `maximise` are provided to find the **global** minimum or maximum, respectively, of a standard Julia function `f` of one or several variables.
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They return an `Interval` that is guaranteed to contain the global minimum (maximum), and a `Vector` of `Interval`s or `IntervalBox`es whose union contains all the minimisers.
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### Examples
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#### 1D
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```
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using IntervalArithmetic, IntervalOptimisation
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julia> @time global_min, minimisers = minimise(x -> (x^2 - 2)^2, -10..11);
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0.046620 seconds (36.07 k allocations: 1.586 MiB)
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julia> global_min
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[0, 1.50881e-09]
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julia> minimisers
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2-element Array{IntervalArithmetic.Interval{Float64},1}:
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[1.41387, 1.41453]
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[-1.41428, -1.41363]
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```
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#### 2D
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```
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julia> @time global_min, minimisers = minimise( X -> ( (x,y) = X; x^2 + y^2 ),
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(-10000..10001) × (-10000..10001) );
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0.051122 seconds (46.80 k allocations: 2.027 MiB)
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julia> global_min
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[0, 2.33167e-08]
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julia> minimisers
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3-element Array{IntervalArithmetic.IntervalBox{2,Float64},1}:
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[-0.000107974, 0.000488103] × [-0.000107974, 0.000488103]
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[-0.000107974, 0.000488103] × [-0.000704051, -0.000107973]
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[-0.000704051, -0.000107973] × [-0.000107974, 0.000488103]
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```
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Note that the last two `IntervalBox`es do not actually contain the global minimum;
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decreasing the tolerance (maximum allowed box diameter) removes them:
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```@autodocs
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Modules = [IntervalOptimisation]
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```
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julia> @time global_min, minimisers = minimise( X -> ( (x,y) = X; x^2 + y^2 ),
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(-10000..10001) × (-10000..10001), 1e-5 );
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0.047196 seconds (50.72 k allocations: 2.180 MiB)
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julia> minimisers
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1-element Array{IntervalArithmetic.IntervalBox{2,Float64},1}:
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[-5.52321e-06, 3.79049e-06] × [-5.52321e-06, 3.79049e-06]
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```
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## Author
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- [David P. Sanders](http://sistemas.fciencias.unam.mx/~dsanders),
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Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México (UNAM)
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## References:
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- *Validated Numerics: A Short Introduction to Rigorous Computations*, W. Tucker, Princeton University Press (2010)
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- *Applied Interval Analysis*, Luc Jaulin, Michel Kieffer, Olivier Didrit, Eric Walter (2001)
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- van Emden M.H., Moa B. (2004). Termination Criteria in the Moore-Skelboe Algorithm for Global Optimization by Interval Arithmetic. In: Floudas C.A., Pardalos P. (eds), *Frontiers in Global Optimization. Nonconvex Optimization and Its Applications*, vol. 74. Springer, Boston, MA. [Preprint](http://webhome.cs.uvic.ca/~vanemden/Publications/mooreSkelb.pdf)
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- H. Ratschek and J. Rokne, [*New Computer Methods for Global Optimization*](http://pages.cpsc.ucalgary.ca/~rokne/global_book.pdf)
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## Acknowledements
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Financial support is acknowledged from DGAPA-UNAM PAPIIT grant IN-117117.

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