cutting_planes(B::Vector{Float64}, alpha::Float64, rel_losses::Array{Float64,2};
               tol::Float64=1e-6, maxiters::Int=1000, ub_w::Float64=2000., debug::Int=0)

Compute the investment weights on the assets in order to build a CV@R-alpha risk budgeting portfolio using a cutting plane method.

This leads to a decomposition of the simulations of relative losses in rel_losses in smaller subproblems, and allow us to scale for several thousand realizations. The portfolio is such that each asset j contributs to the total risk with B[j], which is also called a risk appetite.

The algorithm stops after reaching a provable optimality gap within (both absolute and relative) tolerance tol, or after maxiters iterations, if it fails to converge.

In the beginning, the algorithm needs a bounding box for the weights. Since the logarithm already implies w > 0, we use a sufficiently large bound ub_w for all weights. If the algorithm converges to a solution that has any weight that large, it should be checked for soundness. The most common case is when the problem is unbounded below, for example if alpha is not large enough.

debug >= 1 prints the iteration numbers, gap and change in the weights. debug >= 2 also prints the current upper bound and trial points (weights, V@R)

Returns a triplet (result, weights, V@R-alpha)

cvar_rbp(B::Vector{Float64}, alpha::Float64, rel_losses::Array{Float64,2}; tol::Float64=1e-6, maxiters::Int=1000)

Compute the investment weights on the assets in order to build a CV@R-alpha risk budgeting portfolio given risk appetites in B and a matrix of relative losses in rel_losses. This matrix has size (|B|, nsamples), so that each column corresponds to a sample of relative losses, and each row to a given asset.

The algorithm stops after reaching a provable optimality gap within (both absolute and relative) tolerance tol, or after maxiters iterations, if it fails to converge.

Returns (failed, w) :: Bool, Vector{Float64}

For more details on the algorithm, see cutting_planes.