A simple interior point method implementation. Ref: http://www.maths.ed.ac.uk/~gondzio/reports/mfCS.pdf

ipm.jl

#Problem-specific thing
dualitygap(x...) = nothing
solveeq(x...) = rand()

############################

#Simple backtracking line search
function linesearch(v, dv, α₀= 1.0, δ = 0.6)
    α = α₀
    while v + α*dv ≤ 0
        α *= δ
    end
    return α
end

function myfunc(z, s, σ₁, σ₂, σ₃, α̃, ᾱ)
    while dualitygap() ≥ ϵ do

        if k ≠ 1 && (αP ≤ ᾱ || αD ≤ ᾱ )
            σ = σ₂
        else
            σ = σ₁
        end

        #Predictor step
        Δz̄, Δs̄ = solveeq(σ, z, s)

        αP = linesearch(z, Δz̄)
        αD = linesearch(s, Δs̄)

        #Corrector step
        if (αP ≤ α̃ || αD ≤ α̃ )
            Δz̃, Δs̃ = solveeq(σ₃, z, s)
            Δz = Δz̄ + Δz̃
            Δs = Δs̄ + Δs̃

            αP = linesearch(z, Δz)
            αD = linesearch(s, Δs)

            z = z + αP*Δz
            s = s + αD*Δs
        else
            z = z + αP*Δz̄
            s = s + αD*Δs̄
        end
    end
end