571 lines
88 KiB
Plaintext
571 lines
88 KiB
Plaintext
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 27,
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"id": "cb5170b7-6a4c-415f-8675-8382c5ce837c",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"NWTN (generic function with 1 method)"
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]
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},
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"execution_count": 27,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"using LinearAlgebra, Printf, Plots\n",
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"\n",
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"function NWTN(f;\n",
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" x::Union{Nothing, Vector}=nothing,\n",
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" eps::Real=1e-6,\n",
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" MaxFeval::Integer=1000,\n",
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" m1::Real=1e-4,\n",
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" m2::Real=0.9,\n",
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" delta::Real=1e-6,\n",
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" tau::Real=0.9,\n",
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" sfgrd::Real=0.2,\n",
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" MInf::Real=-Inf,\n",
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" mina::Real=1e-16,\n",
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" plt::Union{Plots.Plot, Nothing}=nothing,\n",
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" plotatend::Bool=true,\n",
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" Plotf::Integer=0,\n",
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" printing::Bool=true)::Tuple{AbstractArray, String}\n",
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"\n",
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" \n",
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" # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
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" # inner functions - - - - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
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" # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
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"\n",
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" function f2phi(alpha, derivate=false)\n",
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" # computes and returns the value of the tomography at alpha\n",
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" #\n",
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" # phi( alpha ) = f( x + alpha * d )\n",
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" #\n",
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" # if Plotf > 2 saves the data in gap() for plotting\n",
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" #\n",
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" # if the second output parameter is required, put there the derivative\n",
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" # of the tomography in alpha\n",
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" #\n",
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" # phi'( alpha ) = < \\nabla f( x + alpha * d ) , d >\n",
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" #\n",
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" # saves the point in lastx, the gradient in lastg, the Hessian in lasth,\n",
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" # and increases feval\n",
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" \n",
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" lastx = x + alpha * d\n",
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" phi, lastg, lastH = f(lastx)\n",
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" \n",
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" if Plotf > 2\n",
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" if fStar > - Inf\n",
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" push!(gap, (phi - fStar) / max(abs(fStar), 1))\n",
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" else\n",
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" push!(gap, phi)\n",
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" end\n",
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" end\n",
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"\n",
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" feval += 1\n",
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"\n",
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" if derivate\n",
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" return (phi, dot(d, lastg))\n",
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" end\n",
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" return (phi, nothing)\n",
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" end\n",
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"\n",
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" # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
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" # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
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" \n",
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" function ArmijoWolfeLS(phi0, phip0, as, m1, m2, tau)\n",
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" # performs an Armijo-Wolfe Line Search.\n",
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" #\n",
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" # phi0 = phi( 0 ), phip0 = phi'( 0 ) < 0\n",
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" #\n",
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" # as > 0 is the first value to be tested: if phi'( as ) < 0 then as is\n",
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" # divided by tau < 1 (hence it is increased) until this does not happen\n",
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" # any longer\n",
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" #\n",
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" # m1 and m2 are the standard Armijo-Wolfe parameters; note that the strong\n",
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" # Wolfe condition is used\n",
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" #\n",
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" # returns the optimal step and the optimal f-value\n",
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" \n",
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" lsiter = 1 # count iterations of first phase\n",
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" local phips, phia\n",
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" while feval ≤ MaxFeval \n",
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" phia, phips = f2phi(as, true)\n",
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" \n",
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" if (phia ≤ phi0 + m1 * as * phip0) && (abs(phips) ≤ - m2 * phip0)\n",
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" if printing\n",
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" @printf(\" %2d\", lsiter)\n",
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" end\n",
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" a = as;\n",
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" return (a, phia) # Armijo + strong Wolfe satisfied, we are done\n",
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" \n",
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" end\n",
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" if phips ≥ 0\n",
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" break\n",
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" end\n",
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" as = as / tau\n",
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" lsiter += 1\n",
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" end \n",
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"\n",
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" if printing\n",
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" @printf(\" %2d \", lsiter)\n",
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" end\n",
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" lsiter = 1 # count iterations of second phase\n",
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" \n",
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" am = 0\n",
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" a = as\n",
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" phipm = phip0\n",
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" while (feval ≤ MaxFeval ) && ((as - am)) > mina && (phips > 1e-12)\n",
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" \n",
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" # compute the new value by safeguarded quadratic interpolation\n",
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" a = (am * phips - as * phipm) / (phips - phipm)\n",
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" a = max(am + ( as - am ) * sfgrd, min(as - ( as - am ) * sfgrd, a))\n",
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" \n",
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" # compute phi(a)\n",
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" phia, phip = f2phi(a, true)\n",
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" \n",
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" if (phia ≤ phi0 + m1 * a * phip0) && (abs(phip) ≤ -m2 * phip0)\n",
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" break # Armijo + strong Wolfe satisfied, we are done\n",
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" end\n",
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" \n",
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" # restrict the interval based on sign of the derivative in a\n",
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" if phip < 0\n",
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" am = a\n",
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" phipm = phip\n",
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" else\n",
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" as = a\n",
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" if as ≤ mina\n",
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" break\n",
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" end\n",
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" phips = phip\n",
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" end\n",
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" lsiter += 1\n",
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" end\n",
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"\n",
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" if printing\n",
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" @printf(\"%2d\", lsiter)\n",
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" end\n",
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" return (a, phia)\n",
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" end\n",
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"\n",
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" # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
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"\n",
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" function BacktrackingLS(phi0, phip0, as, m1, tau)\n",
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" # performs a Backtracking Line Search.\n",
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" #\n",
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" # phi0 = phi( 0 ), phip0 = phi'( 0 ) < 0\n",
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" #\n",
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" # as > 0 is the first value to be tested, which is decreased by\n",
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" # multiplying it by tau < 1 until the Armijo condition with parameter\n",
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" # m1 is satisfied\n",
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" #\n",
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" # returns the optimal step and the optimal f-value\n",
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" \n",
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" lsiter = 1 # count ls iterations\n",
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" while feval ≤ MaxFeval && as > mina\n",
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" phia, _ = f2phi(as)\n",
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" if phia ≤ phi0 + m1 * as * phip0 # Armijo satisfied\n",
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" break # we are done\n",
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" end\n",
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" as *= tau\n",
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" lsiter += 1\n",
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" end\n",
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"\n",
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" if printing\n",
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" @printf(\" %2d\", lsiter)\n",
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" end\n",
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"\n",
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" return (as, phia)\n",
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" end\n",
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" \n",
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" # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
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" # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
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"\n",
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" #Plotf = 2\n",
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" # 0 = nothing is plotted\n",
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" # 1 = the level sets of f and the trajectory are plotted (when n = 2)\n",
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" # 2 = the function value / gap are plotted, iteration-wise\n",
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" # 3 = the function value / gap are plotted, function-evaluation-wise\n",
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"\n",
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" Interactive = false\n",
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"\n",
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" PXY = Matrix{Real}(undef, 2, 0)\n",
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"\n",
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" local gap\n",
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" if Plotf > 1\n",
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" if Plotf == 2\n",
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" MaxIter = 50 # expected number of iterations for the gap plot\n",
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" else\n",
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" MaxIter = 70 # expected number of iterations for the gap plot\n",
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" end\n",
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" gap = []\n",
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" end\n",
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"\n",
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" if Plotf == 2 && plt == nothing\n",
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" plt = plot(xlims=(0, MaxIter), ylims=(1e-15, 1e+1))\n",
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" end\n",
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" if Plotf > 1 && plt == nothing\n",
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" plt = plot(xlims=(0, MaxIter))\n",
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" end\n",
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" if plt == nothing\n",
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" plt = plot()\n",
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" end\n",
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"\n",
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" local fStar\n",
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" if isnothing(x)\n",
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" (fStar, x, _) = f(nothing)\n",
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" else\n",
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" (fStar, _, _) = f(nothing)\n",
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" end\n",
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"\n",
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" n = size(x, 1)\n",
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"\n",
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" if m1 ≤ 0 || m1 ≥ 1\n",
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" throw(ArgumentError(\"m1 is not in (0 ,1)\"))\n",
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" end\n",
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"\n",
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" AWLS = (m2 > 0 && m2 < 1)\n",
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"\n",
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" if delta < 0\n",
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" throw(ArgumentError(\"delta must be ≥ 0\"))\n",
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" end\n",
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"\n",
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" if tau ≤ 0 || tau ≥ 1\n",
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" throw(ArgumentError(\"tau is not in (0 ,1)\"))\n",
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" end\n",
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"\n",
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" if sfgrd ≤ 0 || sfgrd ≥ 1\n",
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" throw(ArgumentError(\"sfgrd is not in (0, 1)\"))\n",
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" end\n",
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"\n",
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" if mina < 0\n",
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" throw(ArgumentError(\"mina must be ≥ 0\"))\n",
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" end\n",
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"\n",
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" # \"global\" variables- - - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
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" # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
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" \n",
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" lastx = zeros(n) # last point visited in the line search\n",
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" lastg = zeros(n) # gradient of lastx\n",
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" lastH = zeros(n, n) # Hessian of lastx\n",
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" d = zeros(n) # Newton's direction\n",
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" feval = 0 # f() evaluations count (\"common\" with LSs)\n",
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"\n",
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" status = \"error\"\n",
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" \n",
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" # initializations - - - - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
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" # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
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"\n",
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" if fStar > -Inf\n",
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" prevv = Inf\n",
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" end\n",
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"\n",
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" if printing\n",
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" @printf(\"Newton's method\\n\")\n",
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" if fStar > -Inf\n",
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" @printf(\"feval\\trel gap\\t\\t|| g(x) ||\\trate\\t\\tdelta\")\n",
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" else\n",
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" @printf(\"feval\\tf(x)\\t\\t\\t|| g(x) ||\\tdelta\")\n",
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" end\n",
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" @printf(\"\\t\\tls it\\ta*\")\n",
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" @printf(\"\\n\\n\")\n",
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" end\n",
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"\n",
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" \n",
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" v, _ = f2phi(0)\n",
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" ng = norm(lastg)\n",
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" if eps < 0\n",
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" ng0 = -ng # norm of first subgradient: why is there a \"-\"? ;-)\n",
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" else\n",
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" ng0 = 1 # un-scaled stopping criterion\n",
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" end\n",
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"\n",
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" # main loop - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
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" # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
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"\n",
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" while true\n",
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"\n",
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" # output statistics - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
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" \n",
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" if fStar > -Inf\n",
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" gapk = ( v - fStar ) / max(abs( fStar ), 1)\n",
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"\n",
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" if printing\n",
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" @printf(\"%4d\\t%1.4e\\t%1.4e\", feval, gapk, ng)\n",
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" if prevv < Inf\n",
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" @printf(\"\\t%1.4e\", ( v - fStar ) / ( prevv - fStar ))\n",
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" else\n",
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" @printf(\"\\t\\t\")\n",
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" end\n",
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" end\n",
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" prevv = v\n",
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" \n",
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" if Plotf > 1\n",
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" if Plotf ≥ 2\n",
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" push!(gap, gapk)\n",
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" end\n",
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" end\n",
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" else\n",
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" if printing\n",
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" @printf(\"%4d\\t%1.8e\\t\\t%1.4e\", feval, v, ng)\n",
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" end\n",
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" \n",
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" if Plotf > 1\n",
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" if Plotf ≥ 2\n",
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" push!(gap, v)\n",
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" end\n",
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" end\n",
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" end\n",
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"\n",
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||
|
|
" # stopping criteria - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
|
||
|
|
"\n",
|
||
|
|
" if ng ≤ eps * ng0\n",
|
||
|
|
" status = \"optimal\"\n",
|
||
|
|
" break\n",
|
||
|
|
" end\n",
|
||
|
|
" \n",
|
||
|
|
" if feval > MaxFeval\n",
|
||
|
|
" status = \"stopped\"\n",
|
||
|
|
" break\n",
|
||
|
|
" end\n",
|
||
|
|
" \n",
|
||
|
|
" # compute Newton's direction- - - - - - - - - - - - - - - - - - - - - -\n",
|
||
|
|
"\n",
|
||
|
|
" lambdan = eigmin(lastH) # smallest eigenvalue\n",
|
||
|
|
" if lambdan < delta\n",
|
||
|
|
" if printing\n",
|
||
|
|
" @printf(\"\\t%1.4e\", delta - lambdan)\n",
|
||
|
|
" end\n",
|
||
|
|
" lastH = lastH + (delta - lambdan) * I\n",
|
||
|
|
" else\n",
|
||
|
|
" if printing\n",
|
||
|
|
" @printf(\"\\t0.00e+00\")\n",
|
||
|
|
" end\n",
|
||
|
|
" end\n",
|
||
|
|
" d = -lastH \\ lastg\n",
|
||
|
|
" phip0 = lastg' * d\n",
|
||
|
|
" \n",
|
||
|
|
" # compute step size - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
|
||
|
|
" # in Newton's method, the default initial stepsize is 1\n",
|
||
|
|
" \n",
|
||
|
|
" if AWLS\n",
|
||
|
|
" a, v = ArmijoWolfeLS(v, phip0, 1, m1, m2, tau)\n",
|
||
|
|
" else\n",
|
||
|
|
" a, v = BacktrackingLS(v, phip0, 1, m1, tau)\n",
|
||
|
|
" end\n",
|
||
|
|
" \n",
|
||
|
|
" # output statistics - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
|
||
|
|
"\n",
|
||
|
|
" if printing\n",
|
||
|
|
" @printf(\"\\t%1.2e\", a)\n",
|
||
|
|
" @printf(\"\\n\")\n",
|
||
|
|
" end\n",
|
||
|
|
" \n",
|
||
|
|
" if a ≤ mina\n",
|
||
|
|
" status = \"error\"\n",
|
||
|
|
" break\n",
|
||
|
|
" end\n",
|
||
|
|
" \n",
|
||
|
|
" if v ≤ MInf\n",
|
||
|
|
" status = \"unbounded\"\n",
|
||
|
|
" break\n",
|
||
|
|
" end\n",
|
||
|
|
" \n",
|
||
|
|
" # compute new point - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
|
||
|
|
" \n",
|
||
|
|
" # possibly plot the trajectory\n",
|
||
|
|
" if n == 2 && Plotf == 1\n",
|
||
|
|
" PXY = hcat(PXY, hcat(x, lastx))\n",
|
||
|
|
" end\n",
|
||
|
|
" \n",
|
||
|
|
" x = lastx\n",
|
||
|
|
" ng = norm(lastg)\n",
|
||
|
|
"\n",
|
||
|
|
" # iterate - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
|
||
|
|
" \n",
|
||
|
|
" if Interactive\n",
|
||
|
|
" readline()\n",
|
||
|
|
" end\n",
|
||
|
|
" end\n",
|
||
|
|
"\n",
|
||
|
|
" if plotatend\n",
|
||
|
|
" if Plotf ≥ 2\n",
|
||
|
|
" plot!(plt, gap)\n",
|
||
|
|
" elseif Plotf == 1 && n == 2\n",
|
||
|
|
" plot!(plt, PXY[1, :], PXY[2, :])\n",
|
||
|
|
" end\n",
|
||
|
|
" display(plt)\n",
|
||
|
|
" end\n",
|
||
|
|
"\n",
|
||
|
|
" # end of main loop- - - - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
|
||
|
|
" # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
|
||
|
|
" return (x, status)\n",
|
||
|
|
"end"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 10,
|
||
|
|
"id": "9fa92a74-19bc-46b3-92de-d39aa74f0075",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [],
|
||
|
|
"source": [
|
||
|
|
"include(\"./TestFunctions/TestFunctions.jl\")\n",
|
||
|
|
"TF = TestFunctions();"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 29,
|
||
|
|
"id": "14e530e5-1b32-4a3e-ac62-b67fd8723f06",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"data": {
|
||
|
|
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|
||
|
|
"image/svg+xml": [
|
||
|
|
"<?xml version=\"1.0\" encoding=\"utf-8\"?>\n",
|
||
|
|
"<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"600\" height=\"400\" viewBox=\"0 0 2400 1600\">\n",
|
||
|
|
"<defs>\n",
|
||
|
|
" <clipPath id=\"clip120\">\n",
|
||
|
|
" <rect x=\"0\" y=\"0\" width=\"2400\" height=\"1600\"/>\n",
|
||
|
|
" </clipPath>\n",
|
||
|
|
"</defs>\n",
|
||
|
|
"<path clip-path=\"url(#clip120)\" d=\"M0 1600 L2400 1600 L2400 0 L0 0 Z\" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n",
|
||
|
|
"<defs>\n",
|
||
|
|
" <clipPath id=\"clip121\">\n",
|
||
|
|
" <rect x=\"480\" y=\"0\" width=\"1681\" height=\"1600\"/>\n",
|
||
|
|
" </clipPath>\n",
|
||
|
|
"</defs>\n",
|
||
|
|
"<path clip-path=\"url(#clip120)\" d=\"M186.274 1486.45 L2352.76 1486.45 L2352.76 47.2441 L186.274 47.2441 Z\" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n",
|
||
|
|
"<defs>\n",
|
||
|
|
" <clipPath id=\"clip122\">\n",
|
||
|
|
" <rect x=\"186\" y=\"47\" width=\"2167\" height=\"1440\"/>\n",
|
||
|
|
" </clipPath>\n",
|
||
|
|
"</defs>\n",
|
||
|
|
"<polyline clip-path=\"url(#clip122)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"247.59,1486.45 247.59,47.2441 \"/>\n",
|
||
|
|
"<polyline clip-path=\"url(#clip122)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"758.552,1486.45 758.552,47.2441 \"/>\n",
|
||
|
|
"<polyline clip-path=\"url(#clip122)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"1269.51,1486.45 1269.51,47.2441 \"/>\n",
|
||
|
|
"<polyline clip-path=\"url(#clip122)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"1780.48,1486.45 1780.48,47.2441 \"/>\n",
|
||
|
|
"<polyline clip-path=\"url(#clip122)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:2; stroke-opacity:0.1; fill:none\" points=\"2291.44,1486.45 2291.44,47.2441 \"/>\n",
|
||
|
|
"<polyline clip-path=\"url(#clip120)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"186.274,1486.45 2352.76,1486.45 \"/>\n",
|
||
|
|
"<polyline clip-path=\"url(#clip120)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"247.59,1486.45 247.59,1467.55 \"/>\n",
|
||
|
|
"<polyline clip-path=\"url(#clip120)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"758.552,1486.45 758.552,1467.55 \"/>\n",
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||
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"<polyline clip-path=\"url(#clip120)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"1269.51,1486.45 1269.51,1467.55 \"/>\n",
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||
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|
"<polyline clip-path=\"url(#clip120)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"1780.48,1486.45 1780.48,1467.55 \"/>\n",
|
||
|
|
"<polyline clip-path=\"url(#clip120)\" style=\"stroke:#000000; stroke-linecap:round; stroke-linejoin:round; stroke-width:4; stroke-opacity:1; fill:none\" points=\"2291.44,1486.45 2291.44,1467.55 \"/>\n",
|
||
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