cmdla/11-09/SDG.jl

using LinearAlgebra, Printf, Plots

function SDG(f;
        x::Union{Nothing, Vector}=nothing,
        astart::Real=1,
        eps::Real=1e-6,
        MaxFeval::Integer=1000,
        m1::Real=1e-3,
        m2::Real=0.9,
        tau::Real=0.9,
        sfgrd::Real=0.01,
        MInf::Real=-Inf,
        mina::Real=1e-16,
        plt::Union{Plots.Plot, Nothing}=nothing,
        plotatend::Bool=true,
        Plotf::Integer=0,
        printing::Bool=true)::Tuple{AbstractArray, String}

    # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    # local functions - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

    function f2phi(alpha, derivate=false)
        lastx = x .- alpha .* g
        (phi, lastg, _) = f(lastx)

        if (Plotf > 2)
            if fStar > -Inf
                push!(gap, (phi - fStar) / max(abs(fStar), 1))
            else
                push!(gap, phi)
            end
        end
        feval += 1

        if derivate
            return phi, dot(-g, lastg)
        end
        return phi, nothing
    end

    # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

    function ArmijoWolfeLS(phi0, phip0, as, m1, m2, tau)
        # performs an Armijo-Wolfe Line Search.
        #
        # Inputs:
        #
        # - phi0 = phi( 0 )
        #
        # - phip0 = phi'( 0 ) (< 0)
        #
        # - as (> 0) is the first value to be tested: if the Armijo condition
        #
        #     phi( as ) <= phi0 + m1 * as * phip0
        #
        #   is satisfied but the Wolfe condition is not, which means that the
        #   derivative in as is still negative, which means that longer steps
        #   might be possible), then as is divided by tau < 1 (hence it is
        #   increased) until this does not happen any longer
        #
        # - m1 (> 0 and < 1, typically small, like 0.01) is the parameter of
        #   the Armijo condition
        #
        # - m2 (> m1 > 0, typically large, like 0.9) is the parameter of the
        #   Wolfe condition
        #
        # - tau (> 0 and < 1) is the increasing coefficient for the first phase
        #   (extrapolation)
        #
        # Outputs:
        #
        # - a is the "optimal" step
        #
        # - phia = phi( a ) (the "optimal" f-value)

        lsiter = 1 # count iterations of first phase
        local phips, phia
        while feval ≤ MaxFeval
            (phia, phips) = f2phi(as, true) # compute phi( a ) and phi'( a )

            if phia > phi0 + m1 * as * phip0  # Armijo not satisfied
                break
            end

            if phips ≥ m2 * phip0             # Wolfe satisfied
                if printing
                    @printf("%2d   ", lsiter)
                end
                a = as
                return (a, phia)              # Armijo + Wolfe satisfied, done
            end
            if phips ≥ 0 # derivative is positive, break
                break
            end
            as = as / tau
            lsiter += 1
        end

        if printing
            @printf("%2d ", lsiter)
        end
        lsiter = 1 # count iterations of second phase

        am = 0
        a = as
        phipm = phip0
        while (feval ≤ MaxFeval) && ((as - am) > abs(mina)) && (abs(phips) > 1e-12)

            if (phipm < 0) && (phips > 0)
                # if the derivative in as is positive and that in am is negative,
                # then compute the new step by safeguarded quadratic interpolation
                a = (am * phips - as * phipm) / (phips - phipm)
                a = max(am + ( as - am ) * sfgrd, min(as - ( as - am ) * sfgrd, a))
            else
                a = (as - am) / 2 # else just use dumb binary search
            end

            phia, phipa = f2phi(a, true) # compute phi( a ) and phi'( a )

            if phia ≤ phi0 + m1 * as * phip0   # Armijo satisfied
                if phipa ≥ m2 * phip0          # Wolfe satisfied
                    break                      # Armijo + Wolfe satisfied, done
                end

                am = a         # Armijo is satisfied but Wolfe is not, i.e., the
                phipm = phipa  # derivative is still negative: move the left
                               # endpoint of the interval to a
            else               # Armijo not satisfied
                as = a         # move the right endpoint of the interval to a
                phips = phipa
            end

            lsiter += 1
        end

        if printing
            @printf("%2d", lsiter)
        end

        return (a, phia)
    end

    # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

    function BacktrackingLS(phi0, phip0, as, m1, tau)
        # performs a Backtracking Line Search.
        #
        # phi0 = phi( 0 ), phip0 = phi'( 0 ) < 0
        #
        # as > 0 is the first value to be tested, which is decreased by
        # multiplying it by tau < 1 until the Armijo condition with parameter
        # m1 is satisfied
        #
        # returns the optimal step and the optimal f-value

        local phia

        lsiter = 1 # count ls iterations
        while feval ≤ MaxFeval && as > mina
            (phia, _) = f2phi(as)
            if phia ≤ phi0 + m1 * as * phip0 # Armijo satisfied
                break                        # we are done
            end
            as = as * tau
            lsiter += 1
        end

        if printing
            @printf("\t%2d", lsiter)
        end
        return (as, phia)
    end

    # Plotf = 1
    # 0 = nothing is plotted
    # 1 = the level sets of f and the trajectory are plotted (when n = 2)
    # 2 = the function value / gap are plotted, iteration-wise
    # 3 = the function value / gap are plotted, function-evaluation-wise
    Interactive = false

    local gap
    PXY = Matrix{Real}(undef, 2, 0)

    status = "error"

    if Plotf > 1
        if Plotf == 2
            MaxIter = 200   # expected number of iterations for the gap plot
        else
            MaxIter = 1000  # expected number of iterations for the gap plot
        end
        gap = []
    end

    if x == nothing
        (fStar, x, _) = f(nothing)
    else
        (fStar, _, _) = f(nothing)
    end

    n = size(x, 1)

    if astart == 0
        throw(ArgumentError("astart must be ≠ 0"))
    end

    if m1 ≤ 0 || m1 ≥ 1
        throw(ArgumentError("m1: ($m1) is not in (0, 1)"))
    end

    AWLS = (m2 > 0 && m2 < 1)

    if tau ≤ 0 || tau ≥ 1
        throw(ArgumentError("tau: ($tau) is not in (0, 1)"))
    end

    if sfgrd ≤ 0 || sfgrd ≥ 1
        throw(ArgumentError("sfgrd: ($sfgrd) is not in (0, 1)"))
    end

    if mina < 0
        throw(ArgumentError("mina: ($mina) must be ≥ 0"))
    end

    if Plotf > 1 && plt == nothing
        plt = plot(xlims=(0, MaxIter))
    elseif plt == nothing
        plt = plot()
    end

    # "global" variables- - - - - - - - - - - - - - - - - - - - - - - - - - - -
    # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

    lastx = zeros(n)    # last point visited in the line search
    lastg = zeros(n)    # gradient of lastx
    feval = 1           # f() evaluations count ("common" with LSs)

    # initializations - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

    if printing
        println("Gradient method")
    end
    if fStar > -Inf
        if printing
            print("feval\trel gap\t\t|| g(x) ||\trate\t")
        end
        prevv = Inf
    else
        if printing
            print("feval\tf(x)\t\t\t|| g(x) ||")
        end
    end
    if astart > 0
        if printing
            print("\tls feval\ta*")
        end
    end
    if printing
        print("\n\n")
    end

    # compute first f-value and gradient in x^0 - - - - - - - - - - - - - - - -

    g = zeros(2, 1)
    v, _ = f2phi(0)
    g = lastg

    # compute norm of the (first) gradient- - - - - - - - - - - - - - - - - - -

    ng = norm(g)
    if eps < 0
        ng0 = -ng # norm of first subgradient: why is there a "-"? ;-)
    else
        ng0 = 1   # un-scaled stopping criterion
    end

    # main loop - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

    while true
        # output statistics & plot gap/f-values - - - - - - - - - - - - - - - -

        if fStar > -Inf
            gapk = (v .- fStar) / max(abs(fStar), 1)

            if printing
                @printf("%4d\t%1.4e\t%1.4e", feval, gapk, ng)
            end
            if prevv < Inf
                if printing
                    @printf("\t%1.4e", (v .- fStar) / (prevv - fStar))
                end
            else
                if printing
                    print("          \t  ")
                end
            end
            prevv = v

            if Plotf > 1
                if Plotf ≥ 2
                    push!(gap, gapk)
                end
                plot!(plt, yscale=:log)
                if Plotf == 2
                    plot!(plt, ylims=(1e-15, 1e+1))
                else
                    plot!(plt, ylims=(1e-15, 1e+4))
                end
            end
        else
            if printing
                @printf("%4d\t%1.8e\t\t%1.4e", feval, v, ng)
            end

            if Plotf ≥ 2
                push!(gap, v)
            end
        end

        # stopping criteria - - - - - - - - - - - - - - - - - - - - - - - - - -

        if ng ≤ (eps * ng0)
            status = "optimal"
            if printing
                print("\n")
            end
            break
        end

        if feval > MaxFeval
            status = "stopped"
            if printing
                print("\n")
            end
            break
        end

        # compute step size - - - - - - - - - - - - - - - - - - - - - - - - - -

        phip0 = -ng * ng

        if astart < 0
            # fixed-step approach
            lastx = x .+ astart .* g
            (v, lastg, _) = f(lastx)
            feval = feval + 1
        else
            # line-search approach, either Armijo-Wolfe or Backtracking

            if AWLS
                a, v = ArmijoWolfeLS(v, phip0, astart, m1, m2, tau)
            else
                a, v = BacktrackingLS(v, phip0, astart, m1, tau)
            end
        end

        # output statistics - - - - - - - - - - - - - - - - - - - - - - - - - -

        if astart > 0
            if printing
                @printf("\t%1.4e\n", a)
            end

            if a ≤ mina
                status = "error"
                if printing
                    print("\n")
                end
                break
            end
        else
            if printing
                print("\n")
            end
        end

        if v ≤ MInf
            status = "unbounded"
            if printing
                print("\n")
            end
            break
        end

        # compute new point - - - - - - - - - - - - - - - - - - - - - - - - - -

        # possibly plot the trajectory
        if n == 2 && Plotf == 1
            PXY = hcat(PXY, hcat(x, lastx))
        end

        x = lastx

        # update gradient - - - - - - - - - - - - - - - - - - - - - - - - - - -

        g = lastg
        ng = norm(g)

        # iterate - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

        if Interactive
            readline()
        end
    end

    if plotatend
        if Plotf ≥ 2
            plot!(plt, gap)
        elseif Plotf == 1 && n == 2
            plot!(plt, PXY[1, :], PXY[2, :])
        end
        display(plt)
    end

    # end of main loop- - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

    return (x, status)
end