backtrack_f2.m

function alpha=backtrack_f2(p,x,f,g,alpha_init)
    alpha=alpha_init;
    ro=0.5;
    c=0.1;
    A = [ 8.403755e-01  -8.880321e-01  1.000928e-01  -5.445289e-01  3.035208e-01  -6.003266e-01  4.899653e-01  7.393631e-01  1.711888e+00  -1.941235e-01  -2.138355e+00  -8.395887e-01  1.354594e+00  -1.072155e+00  9.609539e-01  1.240498e-01  1.436697e+00  -1.960900e+00  -1.976982e-01  -1.207845e+00  2.908008e+00  8.252189e-01  1.378972e+00  -1.058180e+00  -4.686156e-01  -2.724694e-01  1.098425e+00  -2.778719e-01  7.015415e-01  -2.051816e+00  -3.538500e-01  -8.235865e-01  -1.577057e+00  5.079747e-01  2.819841e-01  3.347988e-02  -1.333678e+00  1.127492e+00  3.501794e-01  -2.990660e-01  2.288979e-02  -2.619954e-01  -1.750212e+00  -2.856510e-01  -8.313665e-01  -9.792063e-01  -1.156402e+00  -5.335571e-01  -2.002636e+00  9.642294e-01  5.200601e-01  -2.002785e-02  -3.477109e-02  -7.981636e-01  1.018685e+00  -1.332175e-01  -7.145302e-01  1.351386e+00  -2.247711e-01  -5.890290e-01  -2.937536e-01  -8.479262e-01  -1.120128e+00  2.526000e+00  1.655498e+00  3.075352e-01  -1.257118e+00  -8.654680e-01  -1.765341e-01  7.914161e-01  -1.332004e+00  -2.329867e+00  -1.449097e+00  3.335108e-01  3.913536e-01  4.516794e-01  -1.302847e-01  1.836891e-01  -4.761530e-01  8.620216e-01  -1.361694e+00  4.550296e-01  -8.487094e-01  -3.348869e-01  5.527833e-01  1.039091e+00  -1.117639e+00  1.260659e+00  6.601431e-01  -6.786555e-02  -1.952212e-01  -2.176064e-01  -3.031076e-01  2.304562e-02  5.129036e-02  8.260628e-01  1.526977e+00  4.669144e-01  -2.097133e-01  6.251904e-01  1.832273e-01  -1.029768e+00  9.492218e-01  3.070619e-01  1.351749e-01  5.152463e-01  2.614063e-01  -9.414858e-01  -1.623377e-01  -1.460546e-01  -5.320114e-01  1.682104e+00  -8.757293e-01  -4.838151e-01  -7.120045e-01  -1.174212e+00  -1.922395e-01  -2.740702e-01  1.530073e+00  -2.490247e-01  -1.064213e+00  1.603457e+00  1.234679e+00  -2.296265e-01  -1.506160e+00  -4.446278e-01  -1.559410e-01  2.760683e-01  -2.611636e-01  4.434219e-01  3.918942e-01  -1.250679e+00  -9.479609e-01  -7.411061e-01  -5.078176e-01  -3.205755e-01  1.246904e-02  -3.029177e+00  -4.570146e-01  1.242448e+00  -1.066701e+00  9.337282e-01  3.503210e-01  -2.900576e-02  1.824522e-01  -1.565056e+00  -8.453948e-02  1.603946e+00  9.834777e-02  4.137361e-02  -7.341691e-01  -3.081373e-02  2.323470e-01  4.263876e-01  -3.728087e-01  -2.364546e-01  2.023691e+00  -2.258354e+00  2.229446e+00  3.375637e-01  1.000061e+00  -1.664164e+00  -5.900346e-01  -2.780642e-01  4.227157e-01  -1.670201e+00  4.716343e-01  -1.212847e+00  6.619005e-02  6.523559e-01  3.270600e-01  1.082634e+00  1.006077e+00  -6.509077e-01  2.570562e-01  -9.443778e-01  -1.321789e+00  9.248259e-01  4.984908e-05  -5.491891e-02  9.111273e-01  5.945837e-01  3.502012e-01  1.250251e+00  9.297895e-01  2.397633e-01  -6.903611e-01  -6.515536e-01  1.192102e+00  -1.611830e+00  -2.446194e-02  -1.948847e+00  1.020498e+00  8.617163e-01  1.162083e-03  -7.083721e-02  -2.486284e+00  5.811723e-01  -2.192435e+00  -2.319280e+00  7.993371e-02  -9.484810e-01  4.114906e-01  6.769778e-01  8.577325e-01  -6.911591e-01  4.493776e-01  1.006334e-01  8.260700e-01  5.361571e-01  8.978884e-01  -1.319379e-01  -1.472015e-01  1.007773e+00  -2.123655e+00  -5.045864e-01  -1.270594e+00  -3.825848e-01  6.486793e-01  8.257271e-01  -1.014944e+00  -4.710699e-01  1.370249e-01  -2.918634e-01  3.018186e-01  3.999309e-01  -9.299616e-01  -1.768303e-01  -2.132095e+00  1.145362e+00  -6.290908e-01  -1.203850e+00  -2.539447e-01  -1.428647e+00  -2.085762e-02  -5.606650e-01  2.177779e+00  1.138465e+00  -2.496887e+00  4.413269e-01  -1.398138e+00  -2.550552e-01  1.644041e-01  7.477340e-01  -2.730469e-01  1.576300e+00  -4.809372e-01  3.275121e-01  6.647341e-01  8.518859e-02  8.809528e-01  3.232131e-01  -7.841462e-01  -1.805373e+00  1.858593e+00  -6.045301e-01  1.033597e-01  5.631670e-01  1.135970e-01  -9.047262e-01  -4.677146e-01  -1.248899e-01  1.478958e+00  -8.608157e-01  7.846685e-01  3.086231e-01  -2.338600e-01  -1.056973e+00  -2.841410e-01  -8.669028e-02  -1.469395e+00  1.921822e-01  -8.222933e-01  -9.424059e-02  3.362133e-01  -9.046541e-01  -2.882564e-01  3.500628e-01  -1.835859e+00  1.035976e+00  2.424461e+00  9.594005e-01  -3.157720e-01  4.286227e-01  -1.035985e+00  1.877865e+00  9.407044e-01  7.873458e-01  -8.758743e-01  3.199491e-01  -5.582943e-01  -3.114294e-01  -5.700099e-01  -1.025734e+00  -9.087456e-01  -2.098973e-01  -1.698864e+00  6.076006e-01  -1.177983e-01  6.991603e-01  2.696486e-01  4.942871e-01  -1.483121e+00  -1.020264e+00  -4.469950e-01  1.096586e-01  1.128736e+00  -2.899630e-01  1.261551e+00  4.754248e-01  1.174117e+00  1.269471e-01  -6.568159e-01  -1.481399e+00  1.554890e-01  8.185514e-01  -2.925881e-01  -5.407864e-01  -3.086418e-01  -1.096593e+00  -4.930098e-01  -1.807394e-01  4.584111e-02  -6.378312e-02  6.113352e-01  1.093177e-01  1.814015e+00  3.120238e-01  1.804494e+00  -7.231215e-01  5.265470e-01  -2.602509e-01  6.001425e-01  5.939308e-01  -2.186022e+00  -1.327043e+00  -1.441014e+00  4.018445e-01  1.470201e+00  -3.268142e-01  8.123230e-01  5.455401e-01  -1.051632e+00  3.974670e-01  -7.518947e-01  1.516267e+00  -3.256651e-02  1.636000e+00  -4.250585e-01  5.894334e-01  -6.279123e-02  -2.021959e+00  -9.821315e-01  6.125113e-01  -5.488613e-02  -1.118732e+00  -6.263785e-01  2.495177e-01  -9.930190e-01  9.749502e-01  -6.407095e-01  1.808863e+00  -1.079866e+00  1.991894e-01  -1.521027e+00  -7.236311e-01  -5.932503e-01  4.013363e-01  9.421333e-01  3.004860e-01  -3.730707e-01  8.154885e-01  7.988870e-01  1.202053e-01  5.712476e-01  4.127960e-01  -9.869619e-01  7.595683e-01  -6.572013e-01  -6.039185e-01  1.769468e-01  -3.075035e-01  -1.318204e-01  5.953577e-01  1.046833e+00  -1.979586e-01  3.276782e-01  -2.383015e-01  2.295969e-01  4.399979e-01  -6.168659e-01  2.748368e-01  6.011020e-01  9.230795e-02  1.729841e+00  -6.085574e-01  -7.370598e-01  -1.749879e+00  9.104826e-01  8.670826e-01  -7.989284e-02  8.984760e-01  1.837034e-01  2.907901e-01  1.129447e-01  4.399522e-01  1.016624e-01  2.787335e+00  -1.166665e+00  -1.854299e+00  -1.140681e+00  -1.093343e+00  -4.336093e-01  -1.684699e-01  -2.185336e-01  5.413344e-01  3.892662e-01  7.512290e-01  1.778256e+00  1.223063e+00  -1.283256e+00  -2.328955e+00  9.019315e-01  -1.835639e+00  6.675691e-02  3.547949e-02  2.227168e+00  -6.921425e-02  -5.073231e-01  2.358097e-01  2.458049e-01  7.004521e-02  -6.085805e-01  -1.222593e+00  3.165004e-01  -1.342869e+00  -1.032184e+00  1.331216e+00  -4.189032e-01  -1.403217e-01  8.998223e-01  -3.001110e-01  1.029366e+00  -3.450660e-01  1.012802e+00  6.293346e-01  -2.130151e-01  -8.656973e-01  -1.043108e+00  -2.700688e-01  -4.381414e-01  -4.086743e-01  9.835452e-01  -2.976971e-01  1.143679e+00  -5.316201e-01  9.725657e-01  -5.222505e-01  1.765778e-01  9.707378e-01  -4.139723e-01  -4.382705e-01  2.003391e+00  9.509935e-01  -4.320038e-01  6.489407e-01  -3.600763e-01  7.058850e-01  1.415849e+00  -1.604516e+00  1.028853e+00  1.457968e+00  4.747132e-02  1.746257e+00  1.553875e-01  -1.237120e+00  -2.193494e+00  -3.334071e-01  7.135433e-01  3.174077e-01  4.136104e-01  -5.770856e-01  1.440018e-01  -1.638666e+00  -7.600900e-01  -8.187931e-01  5.197289e-01  -1.416006e-02  -1.155529e+00  -9.524916e-03  -6.898105e-01  -6.666992e-01  8.641494e-01  1.134194e-01  3.983628e-01  8.839699e-01  1.802577e-01  5.508545e-01  6.829643e-01  1.170609e+00  4.758606e-01  1.412233e+00  2.260848e-02  -4.786941e-02  1.701335e+00  -5.097117e-01  -2.854960e-03  9.198671e-01  1.498087e-01  1.404933e+00  1.034122e+00  2.915703e-01  -7.776985e-01  5.666961e-01  -1.382621e+00  2.444747e-01  8.084388e-01  2.130417e-01  8.796772e-01  2.038876e+00  9.239324e-01  2.669174e-01  6.416615e-01  4.254854e-01  -1.314724e+00  -4.164112e-01  1.224688e+00  -4.358421e-02  5.824233e-01  -1.006500e+00  6.451674e-02  6.002919e-01  -1.361515e+00  3.475926e-01  -1.818432e-01  -9.395348e-01  -3.753319e-02  -1.896304e+00  -2.127977e+00  -1.176923e+00  -9.905322e-01  -1.173032e+00  -1.725428e+00  2.882281e-01  -1.594184e+00  1.102188e-01  7.870667e-01  -2.226786e-03  9.310876e-02  -3.781571e-01  -1.482676e+00  -4.381859e-02  9.608252e-01  1.738245e+00  -4.302062e-01  -1.627323e+00  1.663475e-01  3.762659e-01  -2.269505e-01  -1.148912e+00  2.024333e+00  -2.359524e+00  -5.099720e-01  -1.321626e+00  -6.361282e-01  3.178514e-01  1.380480e-01  -7.107351e-01  7.770035e-01  6.223939e-01  6.473809e-01  -4.256317e-01  1.048581e+00  6.607071e-01  2.508772e+00  1.063460e+00  1.156922e+00  5.297883e-02  -1.288386e+00  -3.712212e-01  -7.577919e-01  -5.639689e-01  5.551386e-01  -5.567781e-01  -8.951131e-01  -4.093277e-01  -1.608868e-01  4.093344e-01  -9.526360e-01  3.173175e-01  7.802008e-02  1.324385e+00  -2.131705e-01  -1.344786e-01  -1.171356e+00  -1.385263e+00  3.105083e-01  -2.494891e-01  5.037441e-01  -8.926614e-01  1.908512e+00  1.222307e-01  1.047033e+00  -2.269202e-01  -1.625019e-01  6.900519e-01  5.557568e-01  -1.120255e+00  -1.532693e+00  -1.097868e+00  -1.415773e+00  5.957059e-02  -4.112509e-01  -3.680107e-01  -1.360963e+00  7.795674e-01  4.394111e-01  -8.962248e-02  1.021180e+00  -8.739795e-01  4.147003e-01  3.484412e-01  3.492544e-01  -7.292473e-01  3.268402e-01  -5.148816e-01  -8.964462e-01  -1.203268e+00  1.037816e+00  -8.459442e-01  -1.729138e-01  -1.208652e+00  -2.971268e-01  -3.232038e+00  -1.086959e+00  -1.426436e+00  -1.014451e+00  -2.132672e-01  -3.253478e-01  1.944398e+00  -5.717732e-01  -2.500323e-01  -1.569315e+00  -4.773827e-01  -1.337977e+00  3.029902e-02  8.530868e-01  4.042535e-01  -7.006202e-01  -1.630543e+00  1.460013e+00  2.050043e+00  1.205006e-01  -9.899016e-01  1.197771e+00  -5.926562e-01  -4.698094e-01  8.863774e-01  -1.385220e+00  -1.956754e+00  4.206837e-01  4.007380e-01  9.514216e-02  4.966844e-01  1.082241e+00  9.704478e-01  -5.685696e-01  8.099721e-01  1.732474e-01  -5.055426e-01  -1.193306e+00  6.469709e-01  -3.536226e-01  4.643453e-02  -7.929475e-01  -1.550514e+00  1.715864e-01  -6.213912e-02  1.199028e+00  8.017041e-01  1.053305e+00  -7.488768e-01  -9.363265e-01  -1.269087e+00  4.979806e-01  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-4.740834e-01  -1.517982e+00  9.319403e-01  8.826945e-01  -5.307349e-01  -1.036599e-01  9.944972e-02  -5.090569e-01  6.076575e-01  -2.069370e+00  5.200523e-01  1.384069e+00  -4.287913e-01  -1.045835e+00 ];
    A = reshape(A,[500,100]);
    b = [1.572884e+01 1.314154e+01 1.735027e+01 1.194958e+01 1.289401e+01 1.325886e+01 1.687610e+01 1.765072e+01 1.634437e+01 1.099504e+01 1.667984e+01 1.339503e+01 1.794025e+01 1.381049e+01 1.273983e+01 1.874667e+01 1.987294e+01 1.964573e+01 1.721060e+01 1.563901e+01 1.539055e+01 1.523326e+01 1.400882e+01 1.159432e+01 1.859572e+01 1.000905e+01 1.569533e+01 1.800913e+01 1.039741e+01 1.373954e+01 1.662115e+01 1.390132e+01 1.577357e+01 1.298307e+01 1.736100e+01 1.925789e+01 1.978217e+01 1.025653e+01 1.912732e+01 1.400491e+01 1.642986e+01 1.818115e+01 1.522003e+01 1.010237e+01 1.999332e+01 1.803348e+01 1.708704e+01 1.646889e+01 1.528271e+01 1.356166e+01 1.714360e+01 1.952092e+01 1.782459e+01 1.870849e+01 1.997369e+01 1.905424e+01 1.536492e+01 1.935225e+01 1.890536e+01 1.572992e+01 1.536335e+01 1.527358e+01 1.707328e+01 1.006391e+01 1.012624e+01 1.741594e+01 1.415408e+01 1.641433e+01 1.407223e+01 1.997679e+01 1.422894e+01 1.097317e+01 1.548483e+01 1.352416e+01 1.293642e+01 1.754839e+01 1.352511e+01 1.099030e+01 1.326825e+01 1.779062e+01 1.097979e+01 1.327084e+01 1.772657e+01 1.307204e+01 1.189944e+01 1.925923e+01 1.568739e+01 1.206699e+01 1.200682e+01 1.316090e+01 1.884690e+01 1.719957e+01 1.525841e+01 1.063032e+01 1.054553e+01 1.439234e+01 1.552218e+01 1.030590e+01 1.673268e+01 1.738840e+01 1.941789e+01 1.637770e+01 1.416003e+01 1.252263e+01 1.492113e+01 1.893171e+01 1.649508e+01 1.960099e+01 1.675500e+01 1.789581e+01 1.405270e+01 1.350366e+01 1.302602e+01 1.815829e+01 1.521660e+01 1.201032e+01 1.400681e+01 1.238207e+01 1.398130e+01 1.718417e+01 1.416188e+01 1.616771e+01 1.200393e+01 1.041260e+01 1.556678e+01 1.054609e+01 1.430767e+01 1.423868e+01 1.519191e+01 1.079450e+01 1.828441e+01 1.273172e+01 1.493342e+01 1.101477e+01 1.375442e+01 1.228373e+01 1.484883e+01 1.484376e+01 1.471942e+01 1.953632e+01 1.775202e+01 1.152671e+01 1.535939e+01 1.723130e+01 1.085452e+01 1.120434e+01 1.538817e+01 1.705051e+01 1.367803e+01 1.096554e+01 1.632695e+01 1.510930e+01 1.180604e+01 1.767832e+01 1.320504e+01 1.094114e+01 1.901807e+01 1.705155e+01 1.359489e+01 1.469716e+01 1.195898e+01 1.868499e+01 1.899805e+01 1.968155e+01 1.789397e+01 1.800285e+01 1.001172e+01 1.515442e+01 1.241565e+01 1.001728e+01 1.384662e+01 1.082649e+01 1.510415e+01 1.527108e+01 1.893859e+01 1.432072e+01 1.545921e+01 1.589719e+01 1.930171e+01 1.143647e+01 1.022619e+01 1.164964e+01 1.907337e+01 1.985118e+01 1.249627e+01 1.956572e+01 1.718127e+01 1.932152e+01 1.238908e+01 1.635715e+01 1.613064e+01 1.586217e+01 1.884890e+01 1.600209e+01 1.541888e+01 1.868740e+01 1.428903e+01 1.022100e+01 1.420575e+01 1.558419e+01 1.724813e+01 1.647290e+01 1.287920e+01 1.259981e+01 1.660217e+01 1.003929e+01 1.143387e+01 1.730227e+01 1.592631e+01 1.233081e+01 1.431007e+01 1.443039e+01 1.769543e+01 1.985912e+01 1.721325e+01 1.840578e+01 1.119993e+01 1.191769e+01 1.741980e+01 1.402527e+01 1.423219e+01 1.488371e+01 1.856797e+01 1.706488e+01 1.049529e+01 1.473325e+01 1.843285e+01 1.947034e+01 1.889132e+01 1.944060e+01 1.164258e+01 1.192044e+01 1.432971e+01 1.407354e+01 1.299806e+01 1.094515e+01 1.232583e+01 1.729007e+01 1.722803e+01 1.108242e+01 1.403135e+01 1.974097e+01 1.895468e+01 1.996326e+01 1.398241e+01 1.108559e+01 1.940898e+01 1.380025e+01 1.194276e+01 1.455928e+01 1.114441e+01 1.344683e+01 1.874545e+01 1.205283e+01 1.038497e+01 1.392234e+01 1.012394e+01 1.332067e+01 1.337481e+01 1.065972e+01 1.928291e+01 1.591117e+01 1.716859e+01 1.720261e+01 1.057953e+01 1.760796e+01 1.987963e+01 1.304084e+01 1.072688e+01 1.581297e+01 1.953863e+01 1.984664e+01 1.089009e+01 1.345705e+01 1.488858e+01 1.895635e+01 1.465331e+01 1.928785e+01 1.025816e+01 1.062505e+01 1.188941e+01 1.130912e+01 1.174486e+01 1.381408e+01 1.456699e+01 1.729689e+01 1.840866e+01 1.137814e+01 1.953457e+01 1.680101e+01 1.538607e+01 1.938635e+01 1.420312e+01 1.633995e+01 1.425424e+01 1.652394e+01 1.176486e+01 1.921077e+01 1.370856e+01 1.834556e+01 1.864984e+01 1.884050e+01 1.563406e+01 1.755415e+01 1.030124e+01 1.079338e+01 1.576416e+01 1.356661e+01 1.316161e+01 1.236626e+01 1.246354e+01 1.150629e+01 1.124722e+01 1.563257e+01 1.526225e+01 1.749957e+01 1.160812e+01 1.141215e+01 1.106524e+01 1.935930e+01 1.010988e+01 1.203781e+01 1.641984e+01 1.422981e+01 1.957785e+01 1.283834e+01 1.887315e+01 1.065643e+01 1.843383e+01 1.276532e+01 1.571555e+01 1.649645e+01 1.874722e+01 1.487551e+01 1.186096e+01 1.349258e+01 1.434655e+01 1.604675e+01 1.577194e+01 1.506350e+01 1.248834e+01 1.533745e+01 1.772856e+01 1.551045e+01 1.739739e+01 1.947701e+01 1.795687e+01 1.260729e+01 1.998417e+01 1.320411e+01 1.808417e+01 1.735584e+01 1.722978e+01 1.074384e+01 1.667187e+01 1.429331e+01 1.746410e+01 1.313363e+01 1.866402e+01 1.135855e+01 1.736241e+01 1.574615e+01 1.536065e+01 1.595431e+01 1.740114e+01 1.675999e+01 1.864784e+01 1.132773e+01 1.495882e+01 1.924769e+01 1.207392e+01 1.816601e+01 1.466393e+01 1.225883e+01 1.096323e+01 1.566588e+01 1.482688e+01 1.697958e+01 1.019557e+01 1.716271e+01 1.575177e+01 1.565434e+01 1.994013e+01 1.189249e+01 1.380782e+01 1.636649e+01 1.126221e+01 1.743743e+01 1.784299e+01 1.011181e+01 1.808553e+01 1.949389e+01 1.462982e+01 1.731603e+01 1.302936e+01 1.053948e+01 1.880416e+01 1.614411e+01 1.003423e+01 1.963403e+01 1.644569e+01 1.488862e+01 1.178711e+01 1.388925e+01 1.610385e+01 1.652039e+01 1.853291e+01 1.480308e+01 1.548419e+01 1.012669e+01 1.987721e+01 1.037440e+01 1.194148e+01 1.440747e+01 1.243484e+01 1.662682e+01 1.519871e+01 1.720588e+01 1.104973e+01 1.200643e+01 1.946990e+01 1.949370e+01 1.647537e+01 1.038262e+01 1.123355e+01 1.293974e+01 1.680999e+01 1.644283e+01 1.187650e+01 1.416804e+01 1.782348e+01 1.112292e+01 1.413806e+01 1.776161e+01 1.439151e+01 1.388473e+01 1.049866e+01 1.088878e+01 1.087514e+01 1.175391e+01 1.615052e+01 1.657602e+01 1.139910e+01 1.710692e+01 1.975162e+01 1.152393e+01 1.092873e+01 1.227578e+01 1.088579e+01 1.707418e+01 1.971136e+01 1.394643e+01 1.442859e+01 1.814362e+01 1.953700e+01 1.835894e+01 1.288943e+01 1.824165e+01 1.994043e+01 1.734869e+01 1.205137e+01 1.716177e+01 1.170939e+01 1.284677e+01 1.195661e+01 1.084895e+01 1.247427e+01 1.200827e+01 1.280927e+01 1.954505e+01 1.368116e+01 1.223409e+01 1.786377e+01 1.403642e+01 1.287931e+01 1.421864e+01 1.646210e+01 1.170390e+01 1.873195e+01 1.714075e+01 1.159946e+01 1.087385e+01 1.119231e+01 1.996352e+01 1.613784e+01 1.267378e+01 1.250566e+01 1.749356e+01 1.344392e+01 1.721595e+01 1.769157e+01 1.993899e+01 1.318304e+01 1.473944e+01 1.206782e+01 1.784601e+01 1.382192e+01 1.235153e+01 1.441980e+01 1.032140e+01 ];
    b = b';
    looper=0;
    
    while (f(x+alpha*p)>f(x)+c*alpha*p'*g(x) || looper==0) 
            alpha=alpha*ro;

            for i=1:length(b)
               if b(i)<A(i,:)*(x+alpha*p)
                    looper = 0;
                    break
               else
                   looper=1;
               end
            end
            alpha = ro*alpha;
    end
end