571 lines
88 KiB
Plaintext
Generated
571 lines
88 KiB
Plaintext
Generated
{
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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",
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"\n",
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" if ng ≤ eps * ng0\n",
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" status = \"optimal\"\n",
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" break\n",
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" end\n",
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" \n",
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" if feval > MaxFeval\n",
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" status = \"stopped\"\n",
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" break\n",
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" end\n",
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" \n",
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" # compute Newton's direction- - - - - - - - - - - - - - - - - - - - - -\n",
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"\n",
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" lambdan = eigmin(lastH) # smallest eigenvalue\n",
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" if lambdan < delta\n",
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" if printing\n",
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" @printf(\"\\t%1.4e\", delta - lambdan)\n",
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" end\n",
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" lastH = lastH + (delta - lambdan) * I\n",
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" else\n",
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" if printing\n",
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" @printf(\"\\t0.00e+00\")\n",
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" end\n",
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" end\n",
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" d = -lastH \\ lastg\n",
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" phip0 = lastg' * d\n",
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" \n",
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" # compute step size - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
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" # in Newton's method, the default initial stepsize is 1\n",
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" \n",
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" if AWLS\n",
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" a, v = ArmijoWolfeLS(v, phip0, 1, m1, m2, tau)\n",
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" else\n",
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" a, v = BacktrackingLS(v, phip0, 1, m1, tau)\n",
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" end\n",
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" \n",
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" # output statistics - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
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"\n",
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" if printing\n",
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" @printf(\"\\t%1.2e\", a)\n",
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" @printf(\"\\n\")\n",
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" end\n",
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" \n",
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" if a ≤ mina\n",
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" status = \"error\"\n",
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" break\n",
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" end\n",
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" \n",
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" if v ≤ MInf\n",
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" status = \"unbounded\"\n",
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" break\n",
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" end\n",
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" \n",
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" # compute new point - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
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" \n",
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" # possibly plot the trajectory\n",
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" if n == 2 && Plotf == 1\n",
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" PXY = hcat(PXY, hcat(x, lastx))\n",
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" end\n",
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" \n",
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" x = lastx\n",
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" ng = norm(lastg)\n",
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"\n",
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" # iterate - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
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" \n",
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" if Interactive\n",
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" readline()\n",
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" end\n",
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" end\n",
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"\n",
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" if plotatend\n",
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" if Plotf ≥ 2\n",
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" plot!(plt, gap)\n",
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" elseif Plotf == 1 && n == 2\n",
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" plot!(plt, PXY[1, :], PXY[2, :])\n",
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" end\n",
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" display(plt)\n",
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" end\n",
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"\n",
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" # end of main loop- - - - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
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" # - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -\n",
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" return (x, status)\n",
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"end"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"id": "9fa92a74-19bc-46b3-92de-d39aa74f0075",
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"metadata": {},
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"outputs": [],
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"source": [
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"include(\"./TestFunctions/TestFunctions.jl\")\n",
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"TF = TestFunctions();"
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]
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},
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|
{
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"cell_type": "code",
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"execution_count": 29,
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"id": "14e530e5-1b32-4a3e-ac62-b67fd8723f06",
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"metadata": {},
|
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"outputs": [
|
|
{
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"data": {
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"image/png": 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