first commit with current lessons
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elvis
247
10-04/GMQ.jl
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247
10-04/GMQ.jl
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using LinearAlgebra
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using Printf
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using Plots
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function gmq(Q::Matrix, q::Vector; x::Union{Vector, Nothing}=nothing, fStar::Real=-Inf, alpha::Real=0 , MaxIter::Int=1000 , eps::Real=1e-6, plt::Union{Plots.Plot, Nothing}=nothing, Plotf::Int=2, printing::Bool=true)::Tuple{Vector, String}
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# Plotf
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# 0 = nothing is plotted
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# 1 = the function value / gap are plotted
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# 2 = the level sets of f and the trajectory are plotted (when n = 2)
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Interactive = true # if we pause at every iteration
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Streamlined = true # if the streamlined version of the algorithm, with
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# only one O( n^2 ) operation per iteration, is used
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# reading and checking input- - - - - - - - - - - - - - - - - - - - - - - -
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# - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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if !isreal(Q)
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throw(ArgumentError(Q, "Q not a real matrix"))
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end
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n = size(Q, 1)
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if n <= 1
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throw(ArgumentError(Q, "Q is too small"))
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end
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if n != size(Q, 2)
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throw(ArgumentError(Q, "Q is not square"))
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end
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if !isreal(q)
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throw(ArgumentError(q, "q not a real vector"))
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end
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if size(q, 1) != n
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throw(ArgumentError(q, "q size does not match with Q"))
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end
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if x == nothing
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x = zeros(n, 1)
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end
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if !isreal(x)
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throw(ArgumentError(x, "x not a real vector"))
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end
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if size(x, 1) != n
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throw(ArgumentError(x, "x size does not match with Q"))
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end
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if MaxIter < 1
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throw(ArgumentError(MaxIter, "MaxIter too small"))
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end
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if eps < 0
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throw(ArgumentError(eps, "eps can not be negative"))
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end
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# initializations - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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# - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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if printing
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print("Gradient method for quadratic functions ")
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if alpha == 0
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print("(optimal stepsize)\n")
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else
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print("(fixed stepsize)\n")
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end
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print("iter\tf(x)\t\t\t||g||")
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end
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if fStar > - Inf
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if printing
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print("\t\tgap\t\trate")
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end
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prevf = Inf
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end
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if printing
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if alpha == 0
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print("\t\talpha")
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end
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print("\n\n")
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end
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i = 0;
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if Plotf == 1
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gap = []
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end
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if Streamlined
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g = Q * x + q
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end
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if Plotf == 1 && plt == nothing
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plt = plot(yscale = :log,
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xlims=(0, MaxIter),
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ylims=(1e-15, Inf),
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guidefontsize=16)
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elseif Plotf == 2 && plt == nothing
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plt = plot()
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end
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status = ""
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# main loop - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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# - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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while true
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if !Streamlined
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g = Q * x + q
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end
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ng = norm(g)
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f = dot((g + q)', x) / 2 # 1/2 x^T Q x + q x
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# = 1/2 ( x^T Q x + 2 q x )
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# = 1/2 x^T ( Q x + q + q )
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# = 1/2 ( q + g ) x
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i += 1
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if printing
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@printf("%4d\t%1.8e\t\t%1.4e", i, f, ng)
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end
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if fStar > -Inf
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gapk = (f - fStar)/maximum([abs(fStar), 1])
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if printing
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@printf("\t%1.4e", gapk)
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if prevf < Inf
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@printf("\t%1.4e", (f - fStar)/(prevf - fStar))
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else
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@printf("\t\t")
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end
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end
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prevf = f
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if Plotf == 1
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push!(gap, gapk)
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end
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end
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# stopping criteria - - - - - - - - - - - - - - - - - - - - - - - - - -
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if ng <= eps
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status = "optimal"
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if alpha == 0 && printing
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print("\n")
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end
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break
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end
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if i > MaxIter
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status = "stopped"
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if alpha == 0 && printing
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print("\n")
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end
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break
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end
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# compute step size - - - - - - - - - - - - - - - - - - - - - - - - - -
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# meanwhile, check if f is unbounded below
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# note that if alpha > 0 this is only used for the unboundedness check
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# which is a bit of a waste, but there you go; anyway, in the
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# streamlined version this only costs O( n )
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if Streamlined
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v = Q * g;
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den = dot(g', v)
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else
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den = dot(g', Q * g)
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end
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if den <= 1e-14
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# this is actually two different cases:
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# - g' * Q * g = 0, i.e., f is linear along g, and since the
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# gradient is not zero, it is unbounded below
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#
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# - g' * Q * g < 0, i.e., g is a direction of negative curvature for
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# f, which is then necessarily unbounded below
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if printing
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if alpha == 0
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print("\n")
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end
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@printf("g' * Q * g = %1.4e ==> unbounded\n", den)
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end
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status = "unbounded"
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break
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end
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if alpha > 0
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t = alpha
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else
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t = ng^2 / den # stepsize
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if printing
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@printf("\t%1.2e", t)
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end
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end
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if printing
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print("\n")
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end
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# compute new point - - - - - - - - - - - - - - - - - - - - - - - - - -
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# possibly plot the trajectory
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if n == 2 && Plotf == 2
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PXY = hcat(vec(x), vec(x - t * g))
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plot!(PXY[1,:],
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PXY[2,:],
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linestyle=:solid,
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linewidth=2,
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markershape=:circle,
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seriescolor=colorant"black",
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label="")
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end
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x = x - t * g
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if Streamlined
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g = g - t * v
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end
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if Interactive
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#readline()
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end
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if Plotf != 0
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#IJulia.clear_output(true)
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#display(plt)
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end
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end
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if Plotf == 1
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plot!(plt,
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gap,
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linewidth=2,
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seriescolor=colorant"black")
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display(plt)
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elseif Plotf == 2
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display(plt)
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end
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(vec(x), status)
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end
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