This package is a toolbox for Frank-Wolfe and conditional gradients algorithms.

Frank-Wolfe algorithms were designed to solve optimization problems of the form `f`

is a differentiable convex function and `C`

is a convex and compact set.
They are especially useful when we know how to optimize a linear function over `C`

in an efficient way.

A paper presenting the package with mathematical explanations and numerous examples can be found here:

The most recent release is available via the julia package manager, e.g., with

```
using Pkg
Pkg.add("FrankWolfe")
```

or the master branch:

`Pkg.add(url="https://github.com/ZIB-IOL/FrankWolfe.jl", rev="master")`

Let's say we want to minimize the Euclidian norm over the probability simplex `Δ`

. Using `FrankWolfe.jl`

, this is what the code looks like (in dimension 3):

```
julia> using FrankWolfe
julia> f(p) = sum(abs2, p) # objective function
julia> grad!(storage, p) = storage .= 2p # in-place gradient computation
# # function d ⟼ argmin ⟨p,d⟩ st. p ∈ Δ
julia> lmo = FrankWolfe.ProbabilitySimplexOracle(1.)
julia> p0 = [1., 0., 0.]
julia> p_opt, _ = frank_wolfe(f, grad!, lmo, p0; verbose=true);
Vanilla Frank-Wolfe Algorithm.
MEMORY_MODE: FrankWolfe.InplaceEmphasis() STEPSIZE: Adaptive EPSILON: 1.0e-7 MAXITERATION: 10000 TYPE: Float64
MOMENTUM: nothing GRADIENTTYPE: Nothing
[ Info: In memory_mode memory iterates are written back into x0!
-------------------------------------------------------------------------------------------------
Type Iteration Primal Dual Dual Gap Time It/sec
-------------------------------------------------------------------------------------------------
I 1 1.000000e+00 -1.000000e+00 2.000000e+00 0.000000e+00 Inf
Last 24 3.333333e-01 3.333332e-01 9.488992e-08 1.533181e+00 1.565373e+01
-------------------------------------------------------------------------------------------------
julia> p_opt
3-element Vector{Float64}:
0.33333334349923327
0.33333332783841896
0.3333333286623478
```

Note that active-set based methods like Away Frank-Wolfe and Blended Pairwise Conditional Gradient also include a post processing step. In post-processing all values are recomputed and in particular the dual gap is computed at the current FW vertex, which might be slightly larger than the best dual gap observed as the gap is not monotonic. This is expected behavior.

To explore the content of the package, go to the documentation.

Beyond those presented in the documentation, many more use cases are implemented in the `examples`

folder.
To run them, you will need to activate the test environment, which can be done simply with TestEnv.jl (we recommend you install it in your base Julia).

```
julia> using TestEnv
julia> TestEnv.activate()
"/tmp/jl_Ux8wKE/Project.toml"
# necessary for plotting
julia> include("examples/plot_utils.jl")
julia> include("examples/linear_regression.jl")
...
```

If you need the plotting utilities in your own code, make sure Plots.jl is included in your current project and run:

```
using Plots
using FrankWolfe
include(joinpath(dirname(pathof(FrankWolfe)), "../examples/plot_utils.jl"))
```