Power Cone (WIP)

[tjdiamandis]

Add power cone projection (working) and derivative of projection (some numerical errors).

Derivative of the projection is not working right now. Errors occur in case 4, when the algorithm cannot find a bounding interval for the bisection search. For example, the error can be reproduced in with the random seed below:

Random.seed!(983)
v = randn(3)
α = rand(0.05:0.05:0.95)
s = MOI.PowerCone(α)
dΠ = MOD.projection_gradient_on_set(MOD.DefaultDistance(), v, s)

I have noticed that these errors occur when \alpha is 0.05 or 0.95 and Phi(r) is approximately -r since one of x or y is negative and abs(z) is small.

  x, y, z = v
  α = s.exponent
  Phi_prod(xi,αi,z,r) = (xi + sqrt(xi^2 + 4*αi*r*(abs(z) - r)))
  Phi(r) = 0.5*(Phi_prod(x,α,z,r)^α * Phi_prod(y,1-α,z,r)^(1-α)) - r

Would appreciate any thoughts or ideas. The errors are relatively rare (15 of 200 tests). Some ideas I had:

• Check bounds to see if they are close to 0 (however doing this naively breaks the projection). Note that the solution interval for Phi(r) is (0, abs(z)
• Relatedly, use Newton's method instead of bisection, which will avoid the interval endpoint sign check
• Adjust the numerical tolerances; maybe expand case 3

[matbesancon]

The values to reproduce the first example are:

julia> v = randn(3)
3-element Array{Float64,1}:
  0.7677486332685353
 -1.680097972220894
  0.10222967523066824

julia> α = 0.05

(just writing them down to keep them for later)

[joaquimg]

ok to close in favor of #45 ?