{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Forward-Backward Splitting\n",
    "===================================\n",
    "\n",
    "$\\newcommand{\\dotp}[2]{\\langle #1, #2 \\rangle}$\n",
    "$\\newcommand{\\enscond}[2]{\\lbrace #1, #2 \\rbrace}$\n",
    "$\\newcommand{\\pd}[2]{ \\frac{ \\partial #1}{\\partial #2} }$\n",
    "$\\newcommand{\\umin}[1]{\\underset{#1}{\\min}\\;}$\n",
    "$\\newcommand{\\umax}[1]{\\underset{#1}{\\max}\\;}$\n",
    "$\\newcommand{\\umin}[1]{\\underset{#1}{\\min}\\;}$\n",
    "$\\newcommand{\\uargmin}[1]{\\underset{#1}{\\arg\\min}\\;}$\n",
    "$\\newcommand{\\norm}[1]{\\|#1\\|}$\n",
    "$\\newcommand{\\abs}[1]{\\left|#1\\right|}$\n",
    "$\\newcommand{\\choice}[1]{ \\left\\{  \\begin{array}{l} #1 \\end{array} \\right. }$\n",
    "$\\newcommand{\\pa}[1]{\\left(#1\\right)}$\n",
    "$\\newcommand{\\diag}[1]{{diag}\\left( #1 \\right)}$\n",
    "$\\newcommand{\\qandq}{\\quad\\text{and}\\quad}$\n",
    "$\\newcommand{\\qwhereq}{\\quad\\text{where}\\quad}$\n",
    "$\\newcommand{\\qifq}{ \\quad \\text{if} \\quad }$\n",
    "$\\newcommand{\\qarrq}{ \\quad \\Longrightarrow \\quad }$\n",
    "$\\newcommand{\\ZZ}{\\mathbb{Z}}$\n",
    "$\\newcommand{\\CC}{\\mathbb{C}}$\n",
    "$\\newcommand{\\RR}{\\mathbb{R}}$\n",
    "$\\newcommand{\\EE}{\\mathbb{E}}$\n",
    "$\\newcommand{\\Zz}{\\mathcal{Z}}$\n",
    "$\\newcommand{\\Ww}{\\mathcal{W}}$\n",
    "$\\newcommand{\\Vv}{\\mathcal{V}}$\n",
    "$\\newcommand{\\Nn}{\\mathcal{N}}$\n",
    "$\\newcommand{\\NN}{\\mathcal{N}}$\n",
    "$\\newcommand{\\Hh}{\\mathcal{H}}$\n",
    "$\\newcommand{\\Bb}{\\mathcal{B}}$\n",
    "$\\newcommand{\\Ee}{\\mathcal{E}}$\n",
    "$\\newcommand{\\Cc}{\\mathcal{C}}$\n",
    "$\\newcommand{\\Gg}{\\mathcal{G}}$\n",
    "$\\newcommand{\\Ss}{\\mathcal{S}}$\n",
    "$\\newcommand{\\Pp}{\\mathcal{P}}$\n",
    "$\\newcommand{\\Ff}{\\mathcal{F}}$\n",
    "$\\newcommand{\\Xx}{\\mathcal{X}}$\n",
    "$\\newcommand{\\Mm}{\\mathcal{M}}$\n",
    "$\\newcommand{\\Ii}{\\mathcal{I}}$\n",
    "$\\newcommand{\\Dd}{\\mathcal{D}}$\n",
    "$\\newcommand{\\Ll}{\\mathcal{L}}$\n",
    "$\\newcommand{\\Tt}{\\mathcal{T}}$\n",
    "$\\newcommand{\\si}{\\sigma}$\n",
    "$\\newcommand{\\al}{\\alpha}$\n",
    "$\\newcommand{\\la}{\\lambda}$\n",
    "$\\newcommand{\\ga}{\\gamma}$\n",
    "$\\newcommand{\\Ga}{\\Gamma}$\n",
    "$\\newcommand{\\La}{\\Lambda}$\n",
    "$\\newcommand{\\si}{\\sigma}$\n",
    "$\\newcommand{\\Si}{\\Sigma}$\n",
    "$\\newcommand{\\be}{\\beta}$\n",
    "$\\newcommand{\\de}{\\delta}$\n",
    "$\\newcommand{\\De}{\\Delta}$\n",
    "$\\newcommand{\\phi}{\\varphi}$\n",
    "$\\newcommand{\\th}{\\theta}$\n",
    "$\\newcommand{\\om}{\\omega}$\n",
    "$\\newcommand{\\Om}{\\Omega}$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "from __future__ import division\n",
    "%pylab inline\n",
    "%load_ext autoreload\n",
    "%autoreload 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "This numerical tour presents the Forward-Backward (FB) algorithm to\n",
    "minimize the sum of a smooth and a simple function. It shows an\n",
    "application to sparse deconvolution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Forward-Backward Algorithm\n",
    "--------------------------\n",
    "We consider the problem of minimizing the sum of two functions\n",
    "$$ E^\\star = \\umin{x \\in \\RR^N} E(x) = f(x) + g(x). $$\n",
    "\n",
    "So, we want to find a vector $x^\\star$ solution to the problem, i.e. a minimizer of $E=f+g$.\n",
    "\n",
    "\n",
    "We assume that $f$ is a $C^1$ function with $\\beta$-Lipschitz gradient.\n",
    "\n",
    "\n",
    "\n",
    "We also assume that $g$ is \"simple\", in the sense that one can compute exactly and quickly its proximity operator, which is defined as\n",
    "$$ \\text{prox}_{\\ga g}(x) = \\uargmin{y \\in \\RR^N} \\frac{1}{2}\\norm{x-y}^2 + \\ga g(y). $$\n",
    "for any $\\ga > 0$.\n",
    "\n",
    "\n",
    "The forward-backward algorithm reads, after initializing $x^{(0)} \\in \\RR^N$,\n",
    "$$ x^{(k+1)} = \\text{prox}_{\\ga g}\\pa{ x^{(k)} - \\ga \\nabla f( x^{(k)} )  }. $$\n",
    "\n",
    "\n",
    "If $0 < \\ga < 2/\\beta$, then this scheme converges to a minimizer of\n",
    "$f+g$.\n",
    "\n",
    "Sparse Regularization of Inverse Problems\n",
    "-----------------------------------------\n",
    "We consider a linear inverse problem\n",
    "$$ y = A x^\\sharp + w  \\in \\RR^P$$\n",
    "where $x^\\sharp \\in \\RR^N$ is the (unknown) signal to recover, $w \\in\n",
    "\\RR^P$ is a noise vector, and $A \\in \\RR^{P \\times N}$ models the\n",
    "acquisition device.\n",
    "\n",
    "\n",
    "To recover an estimate of the signal $x^\\sharp$, we consider  basis\n",
    "pursuit denoising, which makes use of the $\\ell^1$ norm as sparsity\n",
    "enforcing penalty:\n",
    "$$ \\umin{x \\in \\RR^N} \\frac{1}{2} \\norm{A x-y}^2 + \\la \\norm{x}_1, $$\n",
    "where the $\\ell^1$ norm is defined as\n",
    "$$ \\norm{x}_1 = \\sum_i \\abs{x_i}. $$\n",
    "\n",
    "\n",
    "The parameter $\\la$ should be set in accordance to the noise level\n",
    "$\\norm{w}$.\n",
    "\n",
    "\n",
    "This minimization problem can be cast in the form of minimizing $f+g$\n",
    "where\n",
    "$$ f(x) = \\frac{1}{2} \\norm{Ax-y}^2\n",
    "\\qandq g(x) = \\la \\norm{x}_1. $$\n",
    "\n",
    "\n",
    "$f$ is smooth; we have\n",
    "$$ \\nabla f(x) = A^* (A x - y),  $$\n",
    "which is $\\beta$-Lipschitz continuous, with\n",
    "$$ \\beta = \\norm{ A^* A }. $$\n",
    "\n",
    "\n",
    "The $\\ell^1$-norm is \"simple\", because its proximal operator is soft\n",
    "thresholding:\n",
    "$$ \\big(\\text{prox}_{\\ga g}(x)\\big)_n = \\max\\pa{ 0, 1 - \\frac{\\la \\ga}{\\abs{x_n}} } x_n. $$\n",
    "\n",
    "Signal Deconvolution on Synthetic Sparse Data\n",
    "-----------------------------------------------------\n",
    "A simple linearized model of seismic acquisition considers a linear filtering\n",
    "operator (convolution):\n",
    "$$ A x = h \\ast x $$\n",
    "\n",
    "The filter $h$ is called the impulse response, or the poind spread function, of the acquisition process $x\\mapsto Ax$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "N = 1024"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We define the width of the filter $h$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "s = 5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We define $h$ as the second derivative of a\n",
    "Gaussian."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "t = arange(-N/2,N/2)\n",
    "h = (1-t**2/s**2)*exp(-(t**2)/(2*s**2))\n",
    "h = h - h.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We define the operator $A$. For simplicity, here periodic boundary conditions are used, so that the convolution is efficiently implemented as a product in Fourier domain."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "h_tf = fft.fft(fft.fftshift(h))\n",
    "opA = lambda u : real(fft.ifft(fft.fft(u) * h_tf))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We display the filter $h$ and its spectrum (amplitude of its Fourier transform)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-100, 100)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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g1S3srouAexQ4OYKTW9jXJcCftrAfaa4YcNLK9gKvamE/rQdceZ7aEdqp4l5F8bNKo2LA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\nASdJGiUDTpI0Sv8fKB+DmkQ0b9MAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1062a3e50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "figsize(7,5)\n",
    "plot(t,h)\n",
    "xlim(-100,100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x108607d10>]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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DKaWbVrpm4NDPAUUeXAK8O6V0Y0R8FHjf4hfMcg4mCqyU0ttW+lxEXEW5i7/s\n4ImI2EXxP+bzF730PIr0fJCF4aLR8325pfmKVjoHEfGDFP+ruKX84TwP+EpEvIFMzsFIRPwTiuGQ\nH1v09KDOwZSWf+3ns/R/k4MSERsowuqTKaXPlk8fioizUkoHy2Hg0V2Xx31fzPu//xuBt0fEFcAm\nYGtEfJK8zsEDFKNNo0uRXUfxH/qDtZyDGibYfhb41+XHFwH3LZtMO5XiF/rdLEy2fxl4A0UlMqjJ\ndsYvuhj8OaC4cv+twK5lz2dzDsack/Xl17un/PqHvOgiKOYhP7Ls+Q8B7y0/fh8nT7af9H0xhD/A\nm4HP5XgOgP8JXFR+vL/8+ms5B3V0bgPwSeDrwFeAfYs+936KSbQ7gL+16PnXlq+/C/jtrk9wzf9Y\n3xoFVk7nAPgm8G3gpvLPf8ztHKxwXv42xYq5u4Bruu5Pg1/nj1DM29y86HvgcmAn8OfAncDnge1r\nfV8M4U8ZWKNVglmdA+BVFBd7voVi9G1bXefAjcOSpLlQy8ZhSZKaZmBJkuaCgSVJmgsGliRpLhhY\nkqS5YGBJkuaCgSVJmgsGliRpLvx/5IZqrPd8ZfMAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1084f45d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(t,fft.fftshift(abs(h_tf)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We generate a synthetic sparse signal $x^\\sharp$, with only a small number of nonzero\n",
    "coefficients."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "random.seed(80) # we set the seed of the random number generator for reproducibility purpose."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "s = round(N*.01) # number of nonzero elements of xsharp\n",
    "sel = random.permutation(N)\n",
    "sel = sel[0:s]   # indices of the nonzero elements of xsharp\n",
    "xsharp = zeros(N)\n",
    "xsharp[sel] = sign(randn(s)) * (1-0.3*rand(s))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "noiselevel = 0.2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Compute the measurements $y=A x^\\sharp + w$ where $w$ is a realization\n",
    "of white Gaussian noise."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "y = opA(xsharp) + noiselevel * randn(N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x108711090>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x108567e10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "figsize(14,5)\n",
    "stem(sel,xsharp[sel])\n",
    "xlim(0,N-1)\n",
    "title('signal $x^\\sharp$')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x108a92110>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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dc1OFq19Yhc2ooWiVc2xEeHpqFEZ1bO4J/CnthM2wUeW2oWgfI8aV1zUI0xiK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ZeO0O33nsSxgRaVBkX4oZRfl4stFTYQF7xV2uTYnEc8X2cT75UTDWVSe3dsAER4lgj/I3tav+Mn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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1085705d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "figsize(14,5)\n",
    "plot(range(N),y)\n",
    "xlim(0,N-1)\n",
    "title('signal $y$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Deconvolution\n",
    "---------------------------\n",
    "We now implement the foward-backward algorithm to recover an estimate of the sparse signal\n",
    "\n",
    "\n",
    "We define the regularization parameter $\\la$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "Lambda = 3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We define the proximity operator of $\\ga g$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "def prox_gamma_g (x, gamma, Lambda) :\n",
    "    return x - x/maximum(abs(x)/(Lambda*gamma),1) # soft-thresholding"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We define the gradient operator of $f$. Note that $A^*=A$ because the filter $h$ is symmetric."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise__ \n",
    "\n",
    "Write the code of the function grad_f."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "grad_f = lambda x : # put your code here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We define the Lipschitz constant $\\beta$ of $\\nabla f$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "85.032184045506753"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "beta = abs(fft.fft(h)).max()**2\n",
    "beta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We define the stepsize $\\ga$, which must be smaller than $2/\\beta$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "gamma = 1.9 / beta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We compute the solution of $\\ell_1$ deconvolution (basis pursuit denoising).\n",
    "We keep track of the energy $E_k=f(x^{(k)})+g(x^{(k)})$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise__ \n",
    "\n",
    "Write the code of the forward-backward iteration."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "nbiter = 2000\n",
    "x = y\n",
    "En_array = zeros(nbiter+1)\n",
    "En_array[0] = norm(opA(x) - y)**2/2 + Lambda*norm(x, ord=1)\n",
    "for iter in range(nbiter):  # iter goes from 0 to nbiter-1\n",
    "    # put your code here\n",
    "    En_array[iter+1] = norm(opA(x) - y)**2/2 + Lambda*norm(x, ord=1)  \n",
    "x_restored = x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We display the result."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x1091d6e50>"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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fMi/OGMHkVvF9NOR5ZVEMeQx9Eru3jBGwG9OaVyY9OzEAAADMjBALAADAYAixAAAADIYQ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\nxv8Px1+i7W32KaAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108c884d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, (subfig1,subfig2) = subplots(1,2,figsize=(16,7)) # one figure with two horizontal subfigures\n",
    "subfig1.stem(xsharp)\n",
    "subfig2.stem(x_restored)\n",
    "subfig1.set_title('$x^\\sharp$')\n",
    "subfig2.set_title('$x_\\mathrm{restored}$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We plot the relative error $(E_k-E^\\star)/(E_0-E^\\star)$ in log-scale with respect to $k$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x10cf32b90>]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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vpTok7lnAEqoQdAWwMpPHihUoSZKktujq4LPdH27wUZtEsDfwQqoQ9GJgaMDGFcAgcLVB\nSJIkqfcYfKQx1EMShoLQADCbalDCd6kC0dJMfl6sQEmSJG0Xg4+0A+qR2QuA5wMvAI4BrqURhL6f\nyd3lKpQkSdJIDD7SBNQXSp1HIwjNB35MFYKGwtAPHZ8tSZJUlsFHaqMIdgaeSxWChsIQ1N2g+rYm\nk81lKpQkSZqaDD5SB9Xjsw+hCkAL6tshwAqqEHQlsCSTnxYrUpIkaQow+EiTLIJZwEk0gtA8YCON\nIHQlcIOHx0mSJLWPwUcqLIJpVIfHDQWh5wF7UF1PaCgILXd6nCRJ0vgZfKQuFMHTqQYlPI8qDB0F\nXEcVgr4PXJnJxnIVSpIk9RaDj9QD6ulxc2l0hRYAD9N6eNw1XlxVkiRpZAYfqQfVQxMOo/XwuNnA\nMhpBaEkm9xcrUpIkqYsYfKQ+EcHeNIYmPA+YA9xG0+FxwHqHJkiSpKnI4CP1qQimA0fTOE9oAbAr\nrUFoZSaPFCtSkiRpkhh8pCkkgtm0Hh73HGAtTRPkHJogSZL6kcFHmsIi2J3qkLj5tA5NuJJGGFqT\nyeZiRUqSJLWBwUfSVvXQhEOpAtBQGDoEWEEjDC3J5J5iRUqSJI2DwUfSmCKYBZxIIwydCNxJa1fo\nB5lsKVakJEnSEzD4SNohEewMHElrV2hf4CoaYWhpJg8WK1KSJGkYg4+kCYvgqTRC0HzgeOBmmq4p\nBNzsKG1JklSKwUdS20UwAziWxsCEBcAMGkHIUdqSJGlSGXwkTYpho7QXAEcA19IUhhylLUmSOsXg\nI6mICHajGqXdHIYepumaQjhKW5IktYnBR1JXaBql3XxNoaFR2kNhyFHakiRpXIoHn4g4AzgPOByY\nm5mrmva9G3gj8DhwVmZ+c5T3MPhIfSiCmTRGaS+gMUq7uSvkKG1JkvSEuiH4HA5sAS4Azh4KPhFx\nBPA5YC7wdOAy4LDM3OYXHIOPNDU0jdJu7goNjdIeCkOO0pYkSdtoV2aYNt4XZua6upDhu04DPp+Z\nm4FbI2I9MI/qFxxJU1AmjwPX1LcLoGWU9nzgr4DjI7aO0h4KQ47SliRJbTHu4DOGA2gNObdTdX4k\naatMfgL8T30bPkr7V4C/AWZEtFxTaIWjtCVJ0niMGXwiYjGw3wi7zsnMRTvwOf7FVtKYMtkELKtv\nH4FtRml/CDgyomWU9pJMbi9TsSRJ6iVjBp/MXDiO99wIzG56fGC9bUQRcV7Tw8HMHBzHZ0rqQ5ls\nAL5Y34aP0v5t4OMRPAItXaGrHaUtSVLviogBYKDt7zvRcdYRcQXw9sxcWT8eGm4wj8Zwg2fnCB/k\ncANJE1GP0n42ja7QfOCZwEpau0KO0pYkqUd1w1S304GPAvsADwCrM/Pl9b5zqMZZPwb8WWZ+Y5T3\nMPhIaqtRRmnfRSMIOUpbkqQeUjz4tIPBR1Kn1aO0j6ARhIZGaS+lEYQcpS1JUpcy+EjSONWjtE+i\nEYSOB26m9QKrjtKWJKkLGHwkqU2aRmkPXWD1ecAMWq8p5ChtSZIKMPhIUgc1jdIeCkNHAtfS1BVy\nlLYkSZ1n8JGkSdQ0SnsoCC2AraO0h8KQo7QlSWozg48kFTRslPZQGGoepb2EapT23cWKlCSpDxh8\nJKnLDBulPZ9qgELzKO0lVKO0Hy9WpCRJPcbgI0ldboRR2vOBp+IobUmStpvBR5J6UAT70nqe0PHA\nLbReYNVR2pIk1Qw+ktQH6lHax9DaFXoSrdcUcpS2JGnKMvhIUp+qR2k3d4WOBK6jqSvkKG1J0lRh\n8JGkKaIepX0CjSA0NEq7uSvkKG1JUl8y+EjSFNU0Sru5KzQ0SnsoDDlKW5LUFww+kqStmkZpD4Wh\n5lHaQ2HIUdqSpJ5j8JEkjWrYKO2hMNQ8SnsJ1SjtB4oVKUnSdjD4SJJ2yLBR2vOpzhtqHqW9BFjv\nKG1JUjcx+EiSJmTYKO2hQLQrrdcUcpS2JKkog48kqe2GjdKeDxyFo7QlSQUZfCRJHRfBrsAcWkdp\nP0rT9DhgdSabihUpSeprBh9J0qSrR2k/i9au0LOBq2kdpX1nsSIlSX3F4CNJ6goRPBmYRyMInQTc\nT2tX6JpMHitWpCSpZxl8JEldKYKdgF+itSt0ELCCRhi6KpN7ihUpSeoZBh9JUs+IYC9aL7A6D7iT\n1q6QF1iVJG2jK4JPRJwBnAccDszLzJX19oXA+4EZwCbgHZl5xQivN/hI0hRUX2D1SFq7Qk+jusDq\nUBhamsn9xYqUJHWFbgk+hwNbgAuAszNzVb39WODOzLwzIo4EvpGZB47weoOPJAnYeoHVk2iEoROA\n22jtCt2YyZZiRUqSJl1XBJ+mYq6gKfgM2xfAPcB+mbl52D6DjyRpRBFMB46mCkJDYWgmVQAaCkPL\nMvlZsSIlSR3XrswwrR3FPIFfB1YODz2SJI0lk83Ayvr2TwAR7E8jCL0PODaCG2kNQ7dkUu4EVklS\nV3rCjk9ELAb2G2HXOZm5qH7OiB2f+jC3/wEWZuYPR3hvOz6SpHGLYBfgWBrnCS0AptMahFZk8kix\nIiVJEzJpHZ/MXDieN46IA4GLgNeNFHqannde08PBzBwcz+dJkqaeTH5BNRBhKXA+QASzaQShvweO\niuA6WsPQBrtCktSdImIAGGj7+7bxHJ+3N011mwV8Gzg3My8e43V2fCRJHRXBrlSDEpq7Qo/RGJhw\nJbC6DlGSpC7TFcMNIuJ04KPAPsADwOrMfHlE/CXwLuCmpqcvzMx7hr3e4CNJmlQRBHAIjSA0n+qC\nq2toCkOZ3FGsSEnSVl0RfCb84QYfSVIXiGAPYC6tYeghWrtC19QDFyRJk8jgI0lSh9RdocNovcDq\nIVQT5obC0JJM7i5WpCRNEQYfSZImUQQzgRNpBKETgbtpCkLAtZk8XqxISepDBh9JkgqKYGfgObR2\nhfYHltMIQ1dlcl+xIiWpDxh8JEnqMhE8BTiJRhCaA9xOa1doXSZbihUpST3G4CNJUpeLYBrwXFq7\nQntRXXdoKAwtzeShYkVKUpcz+EiS1IMieBqNyXELgOOAm2mdIHezF1iVpIrBR5KkPhDBDOBYWrtC\nu9IIQUuA5Zn8vFiRklSQwUeSpD4VwYG0doWeC1xP67lCt9kVkjQVGHwkSZoiIngScAKtYShp7Qqt\nyuTRYkVKUocYfCRJmqLqC6w+g8ahcQuAw4FraL3A6sZiRUpSmxh8JEnSVhHsDsyltSv0MK1doasz\n2VysSEkaB4OPJEkaVd0VejatXaFnAqtoCkOZ/KRYkZK0HQw+kiRph0SwJzCPRhg6CbiX1q7QtZk8\nVqxISRrG4CNJkiYkgp2ozg0aCkLzgQOB5TTC0FWZ/LRYkZKmPIOPJElquwj2Bk6kEYbmARtp7Qpd\nn8mWYkVKmlIMPpIkqeMi2Bk4itYLrO4DLKURhpZm8mCxIiX1NYOPJEkqIoKnUp0fNBSEjgd+SGtX\n6CYvsCqpHQw+kiSpK0QwHTiG1q7QHtTXE6IKQ8szebhYkZJ6lsFHkiR1rQgOoPWaQscA62gNQ7fa\nFZL0RAw+kiSpZ0SwC9Uhcc1doZ1pHBp3JbAyk0eLFSmpKxl8JElSz6ovsHoQrV2hI4BraXSFlgAb\n7ApJU5vBR5Ik9ZUIdgNOoLUr9BitQxNWZfKLYkVKmnTFg09EnAGcR3Xhs7mZuWrY/oOAHwDnZuaH\nRnkPg48kSRpR3RU6hEZXaD7V7x3X0NQVyuT2YkVK6rhuCD6HA1uAC4CzRwg+XwEeB5YZfCRJUjtE\nsDswh9au0KO0doVWZ7KpWJGS2qpdmWHaeF+YmevqQrbZFxGvAm4Bx1ZKkqT2qUdif7u+DXWFnkWj\nI/R64NAI1tDaFfpxmYoldYtxB5/RRMQewDuBk4F3tPv9JUmShtSDD9bXt88ARLAHMI8qCL0BuCCC\nh2l0hJYAV2eyuUjRkooYM/hExGJgvxF2nZOZi0Z52XnA+Zn58xipHSRJktRBmfwMuLy+DXWFDqXR\nFXoT8KwIVtPaFbqzTMWSJsOEp7pFxBU0neMTEd8BZte7Z1GdB/RXmfnxEV6bwHubNg1m5uCECpIk\nSXoCEexJoys0HzgJeIDWrtA1doWkyRcRA8BA06Zzu2KcdR183p6ZK0fYdy7wUGZ+eJTXOtxAkiQV\nF8FOwGG0TpA7GFhFa1foJ6VqlKaqbpjqdjrwUWAfqr+QrM7Mlw97jsFHkiT1pAhmAifSCEInAj+l\ntSu0NpPHihUpTQHFg087GHwkSVKvqLtCh9MIQguoDu9fQWtX6J5iRUp9yOAjSZJUWAR7sW1X6Ce0\ndoWuzeTxYkVKPc7gI0mS1GUi2Bl4Dq0XWD0AWE4jCF2Vyb3FipR6jMFHkiSpB0SwN9XUuKGu0Fzg\nDpoOjwN+YFdIGpnBR5IkqQfVXaEjae0KPQ1YRmtX6L5iRUpdxOAjSZLUJyLYh9au0Bzgdlq7Qtdn\nsqVYkVIhBh9JkqQ+FcE04CgaHaH5VJcQWUojCC3N5P5iRUqTxOAjSZI0hUTwVFq7QicAt9HaFbrB\nrpD6jcFHkiRpCotgOnA0jSA0H5jFtl2hB4sVKbWBwUeSJEktItiP1q7Q8cAttHaFbsyk3C+A0g4y\n+EiSJGlMEcwAjqG1K/Rk4CoaQWhZJg8VK1J6AgYfSZIk7bAI9qcRghYAxwI30doVWm9XSN3C4CNJ\nkqQJq7tCx9HaFdqV1q7Q8kx+VqxITWkGH0mSJHVEBE+ntSt0NHADrV2hW+wKaTIYfCRJkjQpItiF\nalBCc1doOtt2hX5erEj1LYOPJEmSiolgNq0XWD0KuJ7WrtCtdoU0UQYfSZIkdY0IdmXbrtBOVAHo\nyvrrykweKVakepLBR5IkSV0rggAOorUrdARwHa1doR/ZFdJYDD6SJEnqKRHsBpxAa1doC61doVWZ\nPFqsSHUdg48kSZJ6Wt0VOoTWIHQ4sJamrlAmG4oVqeIMPpIkSeo7EewOzKF1nPYvaO0Krc7kF8WK\n1KQy+EiSJKnv1V2hZ9HaFToMWEOjK7QOuC2Th0rVqc4x+EiSJGlKimAPYC6NIHQksDewmuowuRVU\n3aGbHZzQ+4oHn4g4AziP6jjMuZm5qmnf0cAFwJOpTlibm5nbtCMNPpIkSWqHCPamOkTuGOB5VMHo\nceByYCnw5UzuKVehxqsbgs/hVKHmAuDsoeATEdOAlcBvZ+baiNgLeCAzt4zwHgYfSZIktV19iNxh\nwEuAAeA0YBPVOO1rgKuARcB9mTxeqExth3ZlhmnjfWFmrqsLGb7rZcA1mbm2ft59465OkiRJGof6\nELcb6tsnItgJeArwXKrfV08D/gG4L4I1wCCwDPi+h8f1p3EHnzEcCmREXArsC3whMz/Ygc+RJEmS\ntksmW4C7qQ59u3xoewQnUl1o9VXA7wHPiOBGYD3wXeDb9f2fG4h625jBJyIWA/uNsOuczFw0ysum\nA8+nOsbyEeBbEbEyMy8f5fmSJElSEZkspT4HqD48bk+qrtAvAS8E3gEcCHwzghXAxVRdpC2Z/KxM\n1RqPMYNPZi4cx3tuAL6TmT8FiIhLgONpStbNIuK8poeDmTk4js+UJEmSJqTu6DwAfK++faoOQwcA\nJwN/ApxBdVTTrAiuBC6hOmTuEaowZFdogiJigOq8rPa+70THWUfEFcDbM3Nl/XgW8C2qrs9m4OvA\nhzPz6yO81uEGkiRJ6ikRvBTYSBWE9gdeSdVQuA94M/BVL7DaPt0w1e104KPAPlTJeHVmvrze91rg\n3UACX8vMd43yHgYfSZIk9bwIDgLeWN9mU11H6Mf17i3A54BF9blG2gHFg087GHwkSZLUTyLYj+o8\noSOozn2H6ppC76jvX0o1VvtVHha3fQw+kiRJUg+ozxM6juq892dThaAtsDX4fJvqPKIzM7m2SJFd\nzOAjSZIk9aD6mkJDQ8aeRH3aCPAc2Hpu0AbgrcB3M9k06UV2keIXMJUkSZK04+rzfIbCzKYInkQ1\nFGw3Gr+fvx+4DFgdwR31/YXATsAa4F0eKrdj7PhIkiRJXSaCGcDhwNOpDo/7KFUXaHb9lO8DbwPu\nohqlfTBwG3BPvw1Q8FA3SZIkaYqJ4L3AtcCvUQWiOfWunwF7AH8JfAJ4dibLIphJdTjdrsCPe/Gw\nOYOPJEmi3dA7AAAJcklEQVSSNMVF8Baq62e+D/gOsAII4BSqi61+AHhmve2zwPnAg5msj+CAzK0j\nt7uWwUeSJEnSVhE8A/hnqu7PEqrws5nqvKGkmiT38vrpTwHupBq1vR54NJPHJ7vm7WHwkSRJkrRD\nIrgQeMOwzZupLq766/Xo7S3AMZlcM+kFjsDgI0mSJGmHRWydBvcl4NVUY7SXARcATwbeDHwDWNv0\nsgQuAXahGrpwVyZfiOD5wL1U1ym6OJOft79eg48kSZKkHRTBAmB/4OvAgkwui+C3qCbIAdwPzBr2\nsoX1batMog5R91IdOvfmTP6l/fUafCRJkiRNggh+A/jysM2fBH6/6fEq4I+B44GN9bYDMvnExD7b\n4CNJkiRpEkSwG3A28NdUAxT+kOrwt+G/y98H7NW8IXOb5+zgZ7cnM+w00TeQJEmS1N/qc3c+WD+8\nqP66vP56WdNTW0IPQMQ2h80VMa10AZIkSZK6XyaPRtV3uane9FlgHrAIOHmMl14XweYRtm+murDq\nZuBg4CPA6fW+X1AdWnfqROse4qFukiRJkrZLBLtm8kgEuwKPAk9q+hrATOARqulvUAWY0To+FwPH\nAP8OvL7e9tfAp6nC1O7AxyD+vh2ZwY6PJEmSpO2SySPNX2GbryONs75/pPeK4Bf13ZubNl+fya0R\nbKC62OrKiVXc4Dk+kiRJkkpqDkt3jvJ1wgw+kiRJkrqFwUeSJElSX/kg8PfApVQXQV0N/Kje9z1g\nCfBAuz7M4QaSJEmSupbX8ZEkSZKk7WTwkSRJktT3xh18IuKMiLguIh6PiOObtj8pIj4fEddExA8i\n4l3tKVWSJEmSxmciHZ+1VFdW/c6w7a8ByMyjgROAN0fEQRP4HO2giBgoXUO/cm07x7XtHNe2c1zb\nznBdO8e17RzXtvuNO/hk5rrMvHGEXXcAu0fEzlRXW90EPDjez9G4DJQuoI8NlC6gjw2ULqCPDZQu\noI8NlC6gTw2ULqCPDZQuoI8NlC5AY2v7OT6Z+Q2qoHMHcCvwwcwc8WqtkiRJkjQZpo21MyIWA/uN\nsOuczFw0ymt+G9gV2B/YG/huRHwrM3840WIlSZIkaTwmfB2fiLgCODszV9WPPw5cmZmfrR9/Crg0\nM788wmvLXURIkiRJUk9ox3V8xuz47IDmQtYBLwE+GxG7AycB54/0Ii9eKkmSJGkyTGSc9ekRsYEq\n2HwtIr5e77oAmBERa4FlwIWZee3ES5UkSZKk8ZnwoW6SJEmS1O3aPtVte0TEKRGxLiJuioi/KFFD\nL4uI2RFxRX0B2Wsj4qx6+94RsTgiboyIb0bErKbXvLte73UR8bJy1Xe/iNg5IlZHxKL6sevaBhEx\nKyK+EhHX1xc3PtG1bY96ra6LiLUR8bmI2MW1HZ+IuDAi7qqPWhjatsNrGREn1P8/boqIf5js/45u\nNMrafrD+mbAmIi6KiJlN+1zb7TDSujbtOzsitkTE3k3bXNftNNraRsSf1t+310bE3zZtd2230yg/\nD+ZFxLL6d7DlETG3aV971jYzJ/UG7AysBw4GpgNXA8+Z7Dp6+UY1ae/Y+v4ewA3Ac4C/A95Zb/8L\n4AP1/SPqdZ5er/t6YKfS/x3degP+HPhP4Kv1Y9e1Pev678Ab6/vTgJmubVvW9WDgFmCX+vEXgde7\ntuNezxcAxwFrm7btyFoOHUmxDJhX378EOKX0f1vp2yhru3Do+w/4gGvbnnWtt88GLgV+COzturbt\ne/bFwGJgev14X9e2bWs7CPxyff/lwBXtXtsSHZ95wPrMvDUzNwNfAE4rUEfPysw7M/Pq+v7PgOuB\npwOvpPrlkvrrq+r7pwGfz8zNmXkr1TfMvEktukdExIHAqcC/0hja4bpOUP1X3Bdk5oUAmflYZj6A\na9sODwKbgd0iYhqwG/BjXNtxyczvAvcN27wja3liROwPPDkzl9XP+4+m10xZI61tZi7OzC31w6XA\ngfV913Y7jfI9C/Bh4J3DtrmuO2CUtX0L8P76d1gy8+56u2u7A0ZZ2zuo/igKMAvYWN9v29qWCD5P\nBzY0Pb693qZxiIiDqRLzUuBpmXlXvesu4Gn1/QOo1nmIaz6684F3AFuatrmuE3cIcHdEfDoiVkXE\nJ6Oa+ujaTlBm/hT4EPAjqsBzf2YuxrVtpx1dy+HbN+Iab483Uv3FFlzbCYmI04DbM/OaYbtc14k7\nFHhhRFwVEYMRMafe7tpO3LuAD0XEj4APAu+ut7dtbUsEH6cptElE7AH8F/BnmflQ876sen5jrbX/\nH4aJiFcAP8nM1bSOaN/KdR23acDxwMcz83jgYaofcFu5tuMTEc8C3krV/j8A2COqC0lv5dq2z3as\npcYhIt4DbMrMz5WupddFxG7AOcC5zZsLldOPpgF7ZeZJVH8o/VLhevrJp4CzMvMg4G3Ahe3+gBLB\nZyPVcadDZtOa1rQdImI6Vej5TGZeXG++KyL2q/fvD/yk3j58zQ+k0T5UwwLglRHxQ+DzwEsi4jO4\nru1wO9VfH5fXj79CFYTudG0nbA7VRaPvzczHgIuA+bi27bQjPwNur7cfOGy7azyKiPhdqkOMX9u0\n2bUdv2dR/SFkTf3v2YHAyoh4Gq5rO9xO9XOW+t+0LRGxD65tO8zLzP+u73+FxmHYbVvbEsFnBXBo\nRBwcETOAM4GvFqijZ0VEUKXiH2TmR5p2fZXqpGbqrxc3bX9NRMyIiEOo2rTLUIvMPCczZ2fmIcBr\ngMsz83W4rhOWmXcCGyLisHrTycB1wCJc24laB5wUEbvWPxtOBn6Aa9tOO/QzoP5+fzCqyYUBvK7p\nNWoSEadQ/dX8tMx8tGmXaztOmbk2M5+WmYfU/57dDhxfH67puk7cxcBLAOp/02Zk5j24tu2wPiJe\nVN9/CXBjfb99aztZ0xuab1STGm6gOjnp3SVq6OUb8Hyqc1CuBlbXt1OAvYHL6m+UbwKzml5zTr3e\n66gnZngbc41fRGOqm+vanjU9BlgOrKH6a9lM17Zta/tOqiC5lurk++mu7bjX8vNU50ptojof9Q3j\nWUvghPr/x3rgo6X/u7rhNsLavhG4Cbit6d+yj7u2417XXwx9zw7bfwv1VDfXdeJrW/98/Uy9ViuB\nAdd2Qmvb/LN2DtU561cDS4Dj2r22XsBUkiRJUt8rcgFTSZIkSZpMBh9JkiRJfc/gI0mSJKnvGXwk\nSZIk9T2DjyRJkqS+Z/CRJEmS1PcMPpIkSZL6nsFHkiRJUt/7/yrJULI+fTywAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108c92ad0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(log10((En_array[0:1800]-En_array[-1])/(En_array[0]-En_array[-1])))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Over-relaxed Forward-Backward\n",
    "-----------------------------\n",
    "It is possible to introduce a relaxation parameter $\\rho$ with $0 < \\rho < 1$. The over-relaxed foward-backward algorithm initializes $x^{(0)} \\in \\RR^N$,\n",
    "and then iterates, for $k=1,2,\\ldots$\n",
    "$$ z^{(k)} = \\text{prox}_{\\ga g}\\pa{\n",
    "              x^{(k-1)} - \\ga \\nabla f( x^{(k-1)} )  }.\n",
    "$$\n",
    "$$ x^{(k)} = z^{(k)} +\n",
    "      \\rho \\pa{ z^{(k)} - x^{(k-1)} } $$\n",
    "\n",
    "\n",
    "Let us assume $\\gamma=1/\\beta$. Convergence of the iterates $x^{(k)}$ and $z^{(k)}$ to a solution is guaranteed\n",
    "for $ 0 < \\rho < 1/2 $. The weaker property of convergence of $ E(x^{(k)}) $\n",
    "to $E^\\star$ is proved, when $ 1/2\\leq \\rho <1 $."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise__ \n",
    "\n",
    "Write the code of the over-relaxed forward-backward iteration."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "gamma = 1/beta\n",
    "nbiter = 1700\n",
    "rho = 0.95\n",
    "x = y\n",
    "En_array_overrelaxed = zeros(nbiter+1)\n",
    "En_array_overrelaxed[0] = norm(opA(x) - y)**2/2 + Lambda*norm(x, ord=1)\n",
    "for iter in range(nbiter):  \n",
    "    # put your code here\n",
    "    En_array_overrelaxed[iter+1] = norm(opA(x) - y)**2/2 + Lambda*norm(x, ord=1)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1085a78d0>]"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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6L/CBzLz2AOcw+Eiqq5pTaU+++TlmXvokJ17fjxGbJhP7trebNMGptCVJalIN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\nQmsQWggMJ/atZOp1Gznj33d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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10d21ee10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(log10((En_array[0:1800]-En_array[-1])/(En_array[0]-En_array[-1])))\n",
    "plot(log10((En_array_overrelaxed[0:1800]-En_array[-1])/(En_array[0]-En_array[-1])))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "As we see, in this example, over-relaxation does not bring any speedup, because $\\gamma$ is lower than without over-relaxation. There are other setting parameters or other problems, for which over-relaxation does bring a significant speedup. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "FISTA-like Accelerated Forward-Backward Algorithm\n",
    "---------------------------\n",
    "We consider the FISTA algorithm introduced in:\n",
    "\n",
    "\n",
    "A. Beck and M. Teboulle,\n",
    "\"A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems\",\n",
    "_SIAM Journal on Imaging Sciences_, 2009.\n",
    "\n",
    "More precisely, we consider a slightly modified version of FISTA, whose convergence is proved, see\n",
    "A. Chambolle and C. Dossal, \"On the convergence of the iterates of \"FISTA\"\", preprint, 2015."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Given an initial estimate $x^{(0)}$ of the solution and a parameter $a>2$, the algorithm sets $\\gamma=1/\\beta$, sets $z^{(0)}=x^{(0)} \\in \\RR^N$,\n",
    "and iterates, for $k=1,2,\\ldots$\n",
    "$$ x^{(k)} = \\text{prox}_{\\ga g}\\pa{\n",
    "              z^{(k-1)} - \\ga \\nabla f( z^{(k-1)} )  }.\n",
    "$$\n",
    "$$ \\alpha_k=(k-1)/(k+a) $$\n",
    "$$ z^{(k)} = x^{(k)} +\n",
    "      \\alpha_k\n",
    "  \\pa{ x^{(k)} - x^{(k-1)} } $$\n",
    "\n",
    "\n",
    "It is proved that the iterates $x^{(k)}$ converge to a solution $x^\\star$ of the problem. Moreover, \n",
    "the  optimal convergence rate for this class of problems is reached,\n",
    "namely\n",
    "$$ E_k - E^\\star = O(1/k^2), $$\n",
    "whereas the convergence rate for the normal forward-backward is only\n",
    "$O(1/k)$.\n",
    "\n",
    "Note the difference between the over-relaxed forward-backward and the accelerated forward-backward: the later is based on an inertia mechanism, of different nature than over-relaxation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise__\n",
    "\n",
    "Write the code of the accelerated forward-backward iteration."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "gamma = 1/beta\n",
    "nbiter = 1700\n",
    "a = 10\n",
    "x = y\n",
    "En_array_fista = zeros(nbiter+1)\n",
    "En_array_fista[0] = norm(opA(x) - y)**2/2 + Lambda*norm(x, ord=1)\n",
    "for iter in range(nbiter): \n",
    "    # put your code here\n",
    "    En_array_fista[iter+1] = norm(opA(x) - y)**2/2 + Lambda*norm(x, ord=1)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/condatl/anaconda/lib/python2.7/site-packages/IPython/kernel/__main__.py:3: RuntimeWarning: divide by zero encountered in log10\n",
      "  app.launch_new_instance()\n",
      "/Users/condatl/anaconda/lib/python2.7/site-packages/IPython/kernel/__main__.py:3: RuntimeWarning: invalid value encountered in log10\n",
      "  app.launch_new_instance()\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x10d984c90>]"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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oS5HsZOE7QCOofE9oEi13tKT44Q0MfeAwPee3p/me/hibgeeA5+NfW/SekIiI\niOQaBZ9MZXYzcAj3n0RdiuQOM3qR+HhcwcEh9H5uM0On7KL/rGa03TqQvPLDhEFoHuF7Qqs88OPR\nVS0iIiJSfwo+mcrsSuAq3D8YdSmSu8xoBYwjnCA3Ccon0WndEUqmvsKgx8rotrQnBYc7YiwG5gCL\nCB+P01pCIiIiklUUfDKVWTdgLdAN98NRlyNNQ3xowiDCjlAYhprt6cegxzYx7P4D9JzfitZvDMB8\nI7CQcGjCPGCTHo8TERGRTKbgk8nMpgJFwAPAf3A/EHFF0gSZ0RY4i4owlH/4LLov2c2Qqa8z6NFC\n2m/oQd6xfIznqXxXaKkHfiTKukVEREQSKfhkMrM2wMeBy4AS4JO4L4q2KGnqzMgHhpHYFWqztTOD\nZ2xmyNQjdF/SgWZ7u2GsBOZWfHng2yMsW0RERJo4BZ9sEb7zcyvwE8Jpb3qsSDKGGZ155z0hJlK0\nfwx9nn2dkqk76fdUEW239sPKd2HMpXJ63BoNTRAREZHGouCTTcwGAFOAx3G/PupyRE7GjCJgFBUT\n5Oz4JDqtaUHx9JcZ/EgZXVZ2ofBgG4z5VHaFFnjg+6KsW0RERHKXgk+2MesALAC+pTV+JJtUGaU9\niZY7hjLw8Vcpnr6P3s+1ovUbfeJDE+YmfG3U0AQRERFpCAo+2cjsvcAtwHDcy6IuR6QuzGhJOEo7\nDEL5RybRffExih/azqCZ0OHFHuQftfjjcRVfSzzwQ1HWLSIiItlJwScbmRnwDOG7Pv+OuhyRhpAw\nSrviXaFJtNnan0GPbaF4+kF6LGhDi109MFYRhqB5hEMTtkVYtoiIiGSJyIOPhS/t30Q4texMd1+a\nsO8G4PPAceBad3/iJNdoWsEHwOxy4IfAeA06kFxlRhsqR2lPovDAWfR+fi8l096k31MFtN/UGzu+\nr0pXaKUHfizKukVERCTzZELwKQHKgb8A11UEHzMbCtwDnAn0AJ4EBrt7eYprNMXgkwe8CHwO9+ei\nLkekMSSM0o53hcon0XF9FwbP2Mrgh4/SbVl7ivZ3xFhMvCMEzPPAd0ZZt4iIiEQv8uCTUMgckoPP\nDUC5u/8y/nkmcJO7z09xbtMLPgBm1wAX4n5F1KWIRCVhlHYYhprvGkO/OTsofmgnfZ9uwemv9sHK\ntyV0heYB6zw48YcoIiIikrsyOfj8EZjv7nfHP98OPObuU1Kc21SDTytgCzAJ95cirkYkI1QZpT0R\nKzubritpJdHiAAAgAElEQVRaUjz9ZQY9dpzOq7tQcKhVlVHaCz3w/VHWLSIiIunVKMHHzGYBXVPs\nutHjI5lrGHwedfcH0/WbyEpm3wMuAi4ixWOAInLCKO2JtHp9GAOe2EbJQ3vp/XxrWu3og/l6kkdp\nb9YobRERkdyRyR2f7wO4+y/in2cCgbsvSHGuA7GETaXuXlqvgrKFWQHhhLcZuN8cdTki2aDKKO2J\n5B+eRI9Fxyl+6DUGPm50WN+D/KPl2DvvCVWM0j4cZd0iIiJSc2Y2GZicsCnIpODzHXdfEv9cMdxg\nPJXDDQZ6ihs16Y4PgFkPYBHwedxnRl2OSLaJj9IeyDtdofKJtN06gIEzt4ajtBe2ocWu7hgrSR6l\n/VqUdYuIiEjNRd7xMbMrgD8AHYG9wDJ3f298342E46zLgP/n7o+f5BpNO/gAmL0L+A/h+z4boy5H\nJNulHKXd57l9FE/bQf/ZBbTb1Ac7vqfK0ASN0hYREclQkQefhqDgE2f2NeArwCeAc4BLgM3ATbjv\njbI0kWwXH6U9lMqu0CQ6vtCFwY9sZfAjR+i2tEOKUdrzPfC3oqxbREREQgo+ucTMgJ8CVxH+xWsa\ncCHhWkjn4JpaJdKQ4qO0J1ARht4ZpT19F31Lm8dHaW+Pd4UqwtBaD/x4lHWLiIg0RQo+uS4MQ38D\nynH/YtTliOSyhFHa4ZpCVnY2XVa2oHjGKwx69DidV3em8ODpGAuoDEILPPA9UdYtIiLSFCj4NAVm\npwEvAB/EfVHU5Yg0JQmjtMMw1HLHMAbMepXi6Xvp82wrWr/eB/MtVAahecCLGqUtIiLSsBR8mgqz\nq4GP4H5R1KWINGUJo7TDIJR3dBLdl5RRPP01Bs6ETuu6kn+keXyB1YowtNADPxBl3SIiItlOwaep\nMCsiHHTwPtyXR12OiISqjNIOw9Bp2wYwcOYWSh46QM/5p9HyzT4Y60nuCm1SV0hERKTmFHyaknBR\n2CG4/1fUpYjIyVUZpT2R/MMT6Dl/LyXTXmfArDw6vNSLvGMkLLA6D1jsgR+Ksm4REZFMpuDTlJi1\nAzYBQ3HfHnU5IlIzKUZpT6TNy10Y/MgWiqcfoseitjTf3QtjDckLrL4SZd0iIiKZRMGnqTH7M/Aa\n7j+JuhQRqTszOlHxaBxMovDtMfR+7nWGTH2Lfk8V0m5zX+z4oSpdoWUe+JEo6xYREYmKgk9TY3YG\nMBPoi2uFeZFcER+lPZLErlD7Da0onr6F4oePxhdY7YGxnMogNM8DdX9FRKRpUPBpisyeAf6I+wNR\nlyIi6RMfpV3ZFSraN4x+pS8z5MHd4QKrr/Qjr3wPJC2wutIDL4uwbBERkbRQ8GmKzD4GfBX3yVGX\nIiKNJz5KeywVQciOT6LT2mOUPPQqgx4po8vKLhQe7IyxmMogNN8DfyvKukVERBqCgk9TFI623gJc\njPuqiKsRkYgkjNKu7Ao139WfAU9sYsi0ffR5pjWtX++P+eskd4XWeuDHo6tcRESk9hR8miqzm4Au\nuH816lJEJHMkjNIOw5CVTaDbst2UTHuNQY8andZ1I/9IO4wFVAahBR74nijrFhERORUFn6bKrDuw\nhnDIwd6oyxGRzFRllHYYhlq90YWBj21kyLQD9JrbhpZv9sfYSvICqy9qgVUREckkCj5Nmdl9wCLc\nfxN1KSKSPaqM0p5I3tGx9Fi4nSFTX2fgzAI6vNSTvGMtMeZTGYYWeuAHoqxbRESaNgWfpsxsCPAs\ncBbuG6MuR0SyU5VR2mEgOm1bKwbP2MiQqYfosbA9zff0x1hPcldok7pCIiLSWBR8mjqzq4EfAz8D\nngPW4RplKyL1U2WU9kTyDw+n9/NbGDrlLfrPKqLd5r7kHc8jOQgt9sAPRVi2iIjkMAUfAbPJwOeB\n8UAv4C/ADbhWeBeRhmFGC2AclQusTqLtlmMUT99KybSjdF/akaL9fTHWkDBBzgN/Jcq6RUQkdyj4\nSDKzzsDfgGPAR3Avj7giEclB8VHaA0jsChW+PZC+T29iyJTd9JvdnDYvDyCv/DDJXaFlHuiHMiIi\nUnsKPnKicJ2fp4CHNPhARBqLGacRdp7j7wqVT6DD+gMMmfYqg2eU0XV5FwoP9sRYTmUQmueBb4+y\nbhERyQ4KPpKaWV9gMTAJ9/XRFiMiTZEZeUAxiV2hon296f/kBoY+uI8+pS05/bWBmO8heYHVlR7o\nXUUREUmm4CMnZ/Yt4DLgAqL8FywiEmdGO5IWWD0+nk5rdjHkwW0MfsTpvLobBYe7YCymMgjN98Df\nirJuERGJXkYEHzO7ErgJKAHGu/uS+PYLgZuBIuAo8F13n5PifAWfdDArABYBv8P9rqjLERGpKr7A\n6jASu0LNd3Vh4MwNDJm6nz7Pnk6rNwZgvEFyV2itB348uspFRKSxZUrwKQHKCaeJXefuS+PbRwGv\nu/vrZjYMeNzde6Y4X8EnXczOBGYA78Z9PWa9gA8AlwJHgLuAaeoIiUimiC+wOoF3ukJlY+m29HWG\nTH2dQY8aHV/oQf7R9hgLqAxCCzzwPVHWLSIi6ZURwSehmDkkBJ8q+wx4C+jq7seq7FPwSSezzwO/\nIfz+twceAR4GmgPXE3aFvojrp6ciknnMKARGEAahMAy1eqMtgx7dQMm0g/Sa24aWbw3A2EryBLkX\ntcCqiEjuyKbg8xHgane/KMU+BZ90M+sBtAVeTFrg1KwlYQhaifs3I6pORKRWzOhGYhDKOzqKngtf\noeTBHQycWUCHDb3IO9YSYz6VYWihB34gyrpFRKTuGi34mNksoGuKXTe6+4z4MSmDT/wxt4eAC919\nc4prK/hEyawtsAK4GvfHoy5HRKS2zGgGjOKdUdpM4rRXm1H88CZKph6mx8L2NN/TH2M9yV2hTeoK\niYhkh4zv+JhZT8I1ZT7r7vNOcp4DsYRNpe5eWu+CpObMLgH+AAyjyqOIIiLZyIxeVAahieQfHk7v\n57cwdMpO+j9ZRLtNfbHjhiUFocUe+KEo6xYRkZCZTQYmJ2wKMi34fCdhqltb4GnCIqdVc546PpnA\n7Engftz/GnUpIiINzYwWwFgqF1idRNutTvH0zZRMO0a3pR1otq8vxlrii6sC84Et6gqJiEQvIzo+\nZnYFYbegI7AXWObu7zWzHwLfB15KOPxC9+T1GBR8MkQ4AW4qMAD3I1GXIyKSTmYY0I/ErlDBwWL6\nPr2JIQ/uov9TzWiztR955eXwzrtC6gqJiEQkI4JPvW+u4JM5zB4lHG+tro+INDlmtAbOpLIrNJF2\nmw5RMm0rxdOP0W1ZJ4oOVO0KzQO2qiskIpJeCj7SsMzOAf4BlCRNfxMRaYLiXaHBJC6wWnCwH31L\nX2LolN30m92cNi8PiHeFEoPQEnWFREQaloKPNDyzZ4H/w/3eqEsREck0ZrQBzqKyK3QW7TfupmTa\nK5Q8VEbX5Z0pfLsfxhqSH5FTV0hEpB4UfKThmV0K3AyMIso/GCIiWcCMfGAIiV2hwre70W/2Swyd\nsod+s1ty+qsDMFdXSESkHhR8pOGZGbAM+AHuj0RdjohItjGjAzCByq7QODq89AZDHnyV4unldFnZ\nmcKD/eNdocQw9LK6QiIiqSn4SHqYfQy4FjhHXR8RkfoxowA4g+SuUDv6PbWeYQ/spe+cVpz+2iDM\nj5MchJaqKyQiElLwkfQwywdeAL6G+xNRlyMikmvM6ELFGG2YBOWj6bD+FYZOeZXi6dBlZVcKDqsr\nJCISp+Aj6WN2EXAn8FHcn4m6HBGRXGZGETCK5K5QCwY88SLDHthP39LWtN4+GKNqV2iJB344uspF\nRBqHgo+kl9n7gT8C5cBjwD24Px9tUSIiTYMZPUnuCp1Bxxc2MnTKa2FXaFV38o8MwFhNchh6RV0h\nEck1Cj6SfuGwg2HAZcCXgZXANbhvi7QuEZEmxozmwFgSw1DhARj4xHqG3befvk+fTqs3ijHKOPFd\nIXWFRCSrKfhI4zJrDlwPXANci/t9EVckItJkxRdY7cM70+OYBOUldHrhRYbdv53i6Xl0WtODgqMD\ngFUkryukrpCIZBUFH4mG2TjgX8BS4DrgTWAAMDz+1RqYoXeDREQalxmtgDNJ7AoV7T/IwJkvMfy+\nt+nzTBtavjkE4xjqColIFlHwkeiYtQR+DvwXcDqwGVgNrAEOAZ8FFgJfxP1gRFWKiDRp8a7QQJK7\nQv3ptG4tw+57g5KH8ui0rif5xwYSdoXeCUMe+CvRVS4ikkzBR6IXvgPECev9mLUA/gG0Aj6Ie1lj\nlyYiIicy43RgPJVhaAJF+3cx6NENDP/3Qfo825YWO4dhHCW5K7RMXSERiYqCj2Q2s0LCaXDP4X5T\nxNWIiEgKZuQBJVQGoYlQ3pPOq1cz/L4dFD+UT8cXe5Nfpq6QiERGwUcyn1k3YAVwAe4roy5HRERO\nzYz2wFlUhqHxFO1/jcEzNnHGvw/R+7l2NN89PEVXaKkHfiS6ykUkVyn4SHYw+xpwBXDhCY/EiYhI\nxjMjn3B4TeUCq5R3pMuqlQy/9y1KphfQYX0f8o4PIlz2ILEr9Gp0lYtIrlDwkexgVgAsB27AfUbU\n5YiISP2Z0RmYQGVXaAxF+7ZQPH0zZ9x7hN7Pt6fZ3hEYhwlDUMU4bXWFRKTWFHwke5hdDsSAMbiX\nR12OiIg0LDMKgZEkd4Va03XFCs64ZyfF04tov6EfeeUDUVdIRGpJwUeyRzj9bRFwM+5Toi5HRETS\nz4zuJK4pBCNptnc9xQ9tiXeFOlK0f2RCVyhxgpy6QiLyDgUfyS5mlwK/Akao6yMi0vSY0QwYQ3JX\nKJ+uy1cw8l+7KJ7ejHab+mM+iHAwTmJXaFt0lYtI1BR8JLuEXZ95wP/g/u+oyxERkWjFF1jtTXJX\naCjN9qxlyNStnHHvMXrO60TRgVEYB6l8T0hdIZEmRsFHso/ZRcAfgGG4H4+6HBERySxmtATGktwV\nKqPb0lWMvGsXgx9uTrvNAzAfiLpCIk1G5MHHzK4EbiJc+OxMd19aZX9vYC0QuPtvT3INBZ+mJOz6\nPAP8Bfd/RV2OiIhktnhXqB+VXaGJQAnNd6+mZNpWzrinjF7zOlH09ihIeldoPpogJ5IzMiH4lADl\nwF+A61IEn/8Ax4GFCj7yDrPJwD+Bsbi/FXE1IiKSZcxoBYwjuSt0mG7LVjHyzl0MfrgZ7TYPxHww\nsIrkMPSKB1pTTiTbRB58EgqZQ5XgY2YfJPyP0dvAAQUfSWL2S+A84PPA6lMubGpmWvxURERSiXeF\nBpDcFRpEs72rKJm6lRF3V3SFRgPHSF5XaIkHfjiq2kWkZjI2+JhZa+AJ4ALguyj4SFVmecC1wLeB\nLsAe4E3gjfjXLqA10BnoA/Ql7C6uBW4H7tA7QiIicjJmtAbGUxmEJkD523RdsYKR/9xN8fTmtNsy\nEPMSYA3JXaGt6gqJZJZGCT5mNgvommLXje4+I35M1eDzG2CBuz9gZjcB+xV85KTMmgNtgU6Ef9Y6\nAx2A/YRhaAuwFcgDzgR+DJQBH8Z9dwQVi4hIlol3hQaR3BUaQNH+FQyZupURdx2j19xOFB0cAzjJ\nXaHFHvihqGoXkczu+DwD9Irvbkv4k/ofufv/pTjXgVjCplJ3L61XQZLbzPKB/yF8vvt83A9GXJGI\niGQhM07nxK7QXrqsWsGof+ym5KHmtN0yEPOhwDqSu0Kb1RUSSR8L3wmfnLApyKTg8x13X5JiX0DY\n8fndSc5Vx0dqL5wO92/gTdy/HnU5IiKS/czIAwaT3BXqS+GB5QyZtoWRd5bR6/mO8a5QAcnrCi32\nwN+OqnaRXBd5x8fMriBck6UjsBdY5u7vrXKMgo+kh1lbwjUcrsb98ajLERGR3GNGG+AsKoPQWVC+\ni85rlzPqjt0MmdactlsGYD4CeJEwDFUEog3qCok0jMiDT0NQ8JF6MbsU+B1wBu7Hoi5HRERyW7wr\nVEJlEJoE9KLg4BJKHtrCqH+U0efZDhQeGgu0JDkILfLA90VVu0g2U/ARCR95mwU8SIp3yERERNLN\njHac0BViBx3XLWf0HbsY8mAz2m0aiDEa2EhlEJoPvOiBl0dVu0i2UPARATAbDTwKDNCgAxERiZoZ\n+cAQkhZYpTsFhxYz+OHNjL7jGH2e6UDRwbFAO2ABlUFogQe+J6raRTKVgo9IBbMpwFxOMjZdREQk\nSma0ByZQ2RU6E9hOu43LGHP7LobdX9EVGgu8QvI47XUeaO06adoUfEQqmI0gXDR3AK6pOiIiktni\nXaFhJHeFupB3dBGDH97EmL8dpd+cjhQeGkO40PdCKoPQAg98Z1S1i0RBwUckkdkDwALcfxN1KSIi\nIrVlRkeSu0LjgFdps3UpY27byRn3VnSFxgPbSX5XaLUHXhZV7SLppuAjkshsOPAk6vqIiEgOMKMA\nGE5lR2gi0JG8YwsYOHMj4/5ylH6zO8YnyPUEFpMQhjzwHVHVLtLQFHxEqjK7D1iC+69Ost+AkcCF\nhM9XdwSMcEXuB4GniPJ/ECIiItUwozPJXaGxwFZO27aEsX/dyYi7i2i3cRDGWcBOKjtC84EVHmjp\nB8lOCj4iVZkNBeYAA3Hfn7C9PfBF4DNAa2AG4RSd7YTBZ2R8/3rCBVHfaNzCRUREas+MQmAElUFo\nItAWO76AAU9s4Mw/HaH/rI4UHh4L9AOWEYahiq7Q9qhqF6kNBR+RVMz+AnQFrgLaAl8DriYMO38D\nnsdTrJlgVgT8BPgIcB7uLzdWySIiIg3FjK4kd4XGAJtovX0JY257k1F3FtFu4+B4V+gAye8KLfPA\nj0RVu8jJKPiIpBIGmNuAK4GjwF3Ab3HfUsPzvwv8FzAxqWskIiKShcwoInyyIbErdBqUz6ff7PWc\n9b9HGPB4h3hXqBhYRfIjci97oMfAJVoKPiLVMTsNOITXcspN+B7QHfFzr0lHaSIiIlEyoxuVIWgS\nMAp4iRY7FzH2rzsYfUch7TcMxpgAlFMZguYDSzzQECFpXAo+Iuli1hZYDVyF+9NRlyMiIpJO8a7Q\naJK7Qi2gfD695q1j4i2HGfRIRVfoDGAt8DyVnSF1hSStFHxE0snso8B3gfGa9CYiIk2NGT1I7gqN\nAF6kaN9Cxv31Dcb9qTntNhVjTKSyK7Qg/s9F6gpJQ1LwEUknszxgCfAT3KdGXY6IiEiUzGhGOCgh\nsStUGHaF5r7ApN8dZtCjHSg4MpYwJG2ksiu0BHjBAz8eUfmS5RR8RNLN7DLgp8BodX1ERESSmdGL\n5AVWhwPrKDywgNF/386E3zeLrys0FugCPEdlGFrkgYYISc0o+IikWzjoYDXwDdxnR12OiIhIJjOj\nBSd2hfKAebR/aRnn31hG8YxOFBw5k/Cdoo3AQsJH5BYAa9QVklQUfEQag9mXgPfj/oEaHl8A9CUc\npb0D98PpK05ERCRzmWFAb5K7QkOBNRS+vYDRd2znrD/k037DIIzxQHdgMTA3/s+FHvhrEZUvGUTB\nR6QxmLUAtgJn4/5SNcf1Ar4MfIEw9OQD7Qlb+t/GfVX6ixUREclsZrQExpLcFSoH5tFu4zLO/fFh\nhk5pS8GRMcB4wv9PXQzMIXxXaKkHfiCa6iUqCj4ijcXsp0Bb3L+eYp8B3wR+ANwD/Bn3tfF9LYFP\nA/9NuKjqj3G18EVERCrEu0L9SA5CJcAqKJ/H4Ec2cN4Py+m6ciRhECoG1hB2hZYA8zzwDdFUL41F\nwUeksZh1J/yPbF/c9yZszwN+A5wPfAj3jSc5vytwL7AJ+KIGJYiIiJycGa2AcSSP0z4CzKPZngVc\neP3bjLyzDYWHRwHvBvYRDkxYGv9arnHauUXBR6Qxmd0DbMb9B/HPBcDtQH/gStzfOMX5rQh/OnUr\n7n9Nb7EiIiK5I94VGkByV2gwsAIrm8eZf3qdkXe2ovPqXhQePgMYRviY+lLgaeAxYJsWWc1eCj4i\njSns2qwhbLO/CtwNnEbY6anZT5XMSoBngcm4r0lTpSIiIjnPjNbAmVQGoWGE79Yuo+DQGob/ezsT\nfn+cLivGYEwGHFgHzAJWAPM98LeiqV5qK/LgY2ZXAjcRPod5prsvTdg3AvgL4V8My+P7j6S4hoKP\nZA+zrxK+r/Mm4X88P0GKP9enuMYXgG8A43Ava/AaRUREmigz2hM+IjcSOJswGB2H8tn0fn497//y\nJjqtO4fw0bli4GXCIDSL8Ieay9UVykyZEHxKCEPNX4DrKoKPhY8ALQE+5e6rzKwdsNfdy1NcQ8FH\nsovZAGACcF+dgks4DGEBcBPujzZwdSIiIhIXf0RuMHAeMBm4nHBK3BqsbCWj79jO+T/Io+XOszE/\nA+gAzCZ8NG4Z8AqwWWsLRS/y4JNQyBySg8+lwCfc/dM1OFfBR5oes08Tdn0masqbiIhI4zAjjzDc\nnAFcRPjU0nnAbihfQdst87n42wWUPNQD+//t3X28XdOdx/HPN48SIRGNJCQhTVFUkUSUablj0jah\ng84wlLbKa8ZDO4NWq6VevVUd1abVDkanFONhPBVVebXUQymqBA3iIUiTlBCpCkojEclv/ljrusfN\nucm95+x7970n3/frdV5n77Ufzjq/e+4+57f32muxI7Al0J/WZGgZ8FA0x/xy3sH6qycnPseT+mff\nDBgBXBURM9rZ1omPrX9Sb3B3AecTcWnZ1TEzM1ufSexGGmj1AGAXUsLzNDCP9908l6ZvBqP/sD19\nVvZB7Av8AriHNFbfX6I5/lRW3dcX3ZL4SLoVGFVl0SkRMTOv0zbx+TLweVIbyzeB24FTI+I3XfUm\nzHqddGX0dNK9Pms/+5Cai04ndZs9EFgJzAJuJ+LpLq6pmZnZeiM3j9uYdFVoW1J32XsDY4Bb2GTe\n4xx4aDDykfH0XTkZxcbAU8AzpC61bwKWRnO8Ws47aEw9+YrPwcD0iPhcnj8VWB4R36+ybQCnVRTd\nGRF31lUhs94gXfV5CjiciHvXst4WwKOktsY3AUuA95Bu3PxH4Bgiru3y+pqZma2ncjK0OTAV+Hdg\nKKlV0zA2eOU+dr54LnuecS+Dlh4AMRmxGfAaaQy/84E+wOxoXvN+d6tOUhPpvqwWzT0p8flyRDyU\n54eRrvJ8mHRm+ibgrIi4qcq2vuJj66/ULHR3Ig5pZ/l44GrgRiK+XWX5JGAmcAYR53ZhTc3MzKyC\nxD8Az5MSodHAfkA/tOoVPnDlaUw//kkGL50GfJp0X1Ef0phCV+XtfgX0ieZYWcob6GVKv+Ij6ZPA\n2aSzz68BsyNiel52GHAyqc/0X0bE19rZhxMfW39JQ4EFwI5EPN9m2XBS2+G7gWPb7QRB2hJ4APgs\nETd3aX3NzMysXRLjgCPzYyxp4PIX6POWGH/nxky8YC7bX7cxip2AnfNmlwDXk8Yh+q6vClVXeuJT\nBCc+tt6TzgNeWOOKjnQG6bL6ER24B2gqcAswlSr30pmZmVn3kRhFuk9oe1KvcJDGFPpKnr6Z8bcN\n4tP7HEXflV8BDiZ1prA5sJzWq0IvA1d7bCEnPmaNQfoEcDwRH60ou5vUVHQiEbM7uJ+jgMOI2Ksr\nqmlmZma1y/cJ7QJMBN5HSoJWw+qg/zJYNeAuhi0cx9997xtMuvAYUhK0gtTldj/g9bz9JODGaI7l\npbyRkjjxMWsE0qakAdL2JOJBpIGksz1HEXFBJ/bTD3gR+F8ivtwldTUzM7NC5DGF+uXZDci3jQDb\nAStgNSieY9w93+CIpuuBs4AvkHp3/RvwP8C+pPuGvgRMiOaY081vo9s48TFrFNJbpEvhA4ETgL2I\n2LeG/byP1M31ZkS8XWgdzczMrMtI7wxXMZjWhOg7wDFo1Wyi72I2WnQ3y0bsxYgnNmTaCcPY8u5r\nUHyG9BtiPKlDsQAGAa8An43m+Fv3v5viOfExaxTSKFLvLgNJ7YHfS8SCGvf1IHAOEZd0cP3hpHbI\nI0idKfQHdibikZpe38zMzAohMYDU1G0LUvO4s0mtRMbmVX7HDld9m93OGcHqfot5Y9QObPnbTRiy\n5ATEamCTvN61QF/guGiORd38NgrhxMeskUh7AP9JGtfn2Tr28yngCmAAsY4uMqVtSGMJATwIDAfe\nm+cnEDG/5nqYmZlZl5A4DXgM+CdSQjQ5L3oDGAKcyhfH/Zkhi79H37eXEoxGDAJg2fAmBi9dFc1x\nTxl1r5UTHzOrTnocOJ2Iq9pZvhGwO3Au8AJwCREX52UfAS4G7gfOA+5dZ69yZmZmVhqJY0mdIp0O\n3EU6mSlgGnAQfVaeyd6nTmDVQNjr9LTRkh2/Huc9eobE5hG8UFLVO8yJj5lVJ30T+BwwvmrSIp1I\nukHyFtIYQdFm+SjSTZN7AJ8C7lrn1SMzMzMrncSWpO/wN4Dfk5KflaT7hoIDDt+GxZPGM+Y+uO6K\nTUkdI+0KzAOWR1B93MCSOfExs+qkDYA3SeMDbVFl+YXALCJ+so79nEJqfgfQl/CgamZmZr2dxEXA\nEW2KVwIzI/jn3PX2amCnCB7t9gpW4cTHzNonpX/slv8vSaSD3C+B+cA+RPx2Hft4P/BknjsL+CYR\nr3dNhc3MzKy7SLQkANcA/0LqRnsW8BNgI+Bo4NdAZRfZQWtnTO8HlkRwlcSHSYOt7gLcEMGy4uvr\nxMfM2iMdSDqY9SUikIYCr5IGTDsc2JXowOBn0seA/YHPA/9GxE+7rtJmZmbWHST2AEaTusDeI4Lb\nJA4l9SAH6TfDsDabfTQ/3hGBchL1MrApcHQE5xdf32Jyhn7rXsXMep2Ia5GWApsBS4CReckM4OgO\nJT1pP7cgTcxz7+/w60ubAQeQDqpjgFWkUae/Qeo6u4mISzu8PzMzMytMBPdWzN6Wy65Y2zYSC2iT\n+EjvJDmb5uejJR4FJgLP57LNI/hx3ZUugK/4mDUq6T7gRCJ+h7QnaXRngGFEvNaJ/QwGTgJ2J+Lj\nHdym5f6g5cBxwDGkg2ATcDCpUwX/75uZmfUSEoOBE4FvkTpQOIbU/K3t9/krtI4hBKQrQ/W9djE5\nQ9ZirqwAAAnRSURBVJ96d2BmPdY8Uv/+AKPy8+udSnoAIpYBl9KZKz6wQX5+gYgLgD/m+fG0HCCl\nvp2qh5mZmZUm37szI89en58fyM+3Vaz6rqQHQFqj2Vwp3NTNrHE9CfwIqRn4TS67ucZ9/QkYhnQH\nKYk6HDgIeJCIqwGQdgJ+Tjqh0nLJe15+fjivfz6p5xiAhUiPETG9xjqZmZlZN4pgudLpy2dy0eXA\nFGAmMHUtmz4uUW1ojJXAoPy8FfAj4JN52QrgZ8A+9da7hZu6mTUqqT/pJsVzgF+QenR7iYi3a9zf\nCNKAp/1InSSkwU4jbsjLDwUOITVtC+CvwAoiliH1AY4CfgwcS+p4YTNSQjTIg6SamZn1DhKDInhT\nYhCpSfsGFc8ChpKG1RiYN1nBmh0ltLgB2Am4hHRSFVJTuotJydSGwH+Dvu/ODcysfWnQ0YVIC4AN\niVhc5/5eQnqZ1FFCkK78vFixxkhgPhELq2y7Gqml/AkilgJLkd4CNgY61/zOzMzMShHBm5XPsMZz\nte6sX622L4kVefKPFcVPRrBQ4jlgOvBQfTVu5Xt8zBrfi7Te41OvN/LzENZMfNrOV6sHbdYpsm5m\nZmbWO1UmS21/L6ztt0WnOPExa3xFJhctHRJMBMaRusoGaQqwM2s/OC2pqE9l3aYiTUHaoso2IG2Z\n7x8CaWxed1OkiUiT26y7LdK2nXpHayMNz683cN0r9xDSe5F2LLsa1g2kHZAmdGL9CUg7VCmfglT+\nCYii/39t/SNth7R12dWwujnxMbOa/ZE0sFgRzgfmA7sCvyai5bL2ucAH8rL2vJTr8npF2f3AyXn7\n9jo5OBA4LU/vl9edCHyX1t5lWnwe+EJH3kgHfTC/3ho91PRgh5Fiao3vJFrbxHfE4Xmbts4ldTVf\ntqL/f23980XgX8uuhHXKDOD7pM6XXgZmA8/mZfcAv6fA5vDu3MDMzMzMzHosj+NjZmZmZmbWQU58\nzMzMzMys4dWc+Eg6SNLjklZJmlhRvoGkKyU9KukJSV8rpqpmZmZmZma1qeeKzxzSyKp3tSk/BCAi\nPghMAo6WNK6O17FOktRUdh0alWPbdRzbruPYdh3Htms4rl3Hse06jm3PV3PiExFzI+LpKosWAxtK\n6ksabfUt0gju1n2ayq5AA2squwINrKnsCjSwprIr0MCayq5Ag2oquwINrKnsCjSwprIrYGtX+D0+\nEfFrUqKzGFgIzIiIqqO1mpmZmZmZdYd+a1so6VaqD3x4SkTMbGebTwODgNHAcOBuSbdHxIJ6K2tm\nZmZmZlaLusfxkXQHcGJE/CHPnwfcGxGX5/kLgZsj4mdVti1vECEzMzMzM+sVihjHZ61XfDqhsiJz\ngb2ByyVtCHwI+GG1jTx4qZmZmZmZdYd6urP+pKTnSInNLyXdlBf9BBggaQ4wC7goIh6rv6pmZmZm\nZma1qbupm5mZmZmZWU9XeK9uHSFpmqS5kp6R9NUy6tCbSRor6Y48gOxjko7L5cMl3SrpaUm3SBpW\nsc3JOd5zJX2svNr3fJL6SpotaWaed1wLIGmYpGslPZkHN97NsS1GjtXjkuZIukLSQMe2NpIukrQk\nt1poKet0LCVNyn+PZyT9V3e/j56ondjOyMeERyRdL2loxTLHtgOqxbVi2YmSVksaXlHmuHZQe7GV\n9B/5c/uYpO9WlDu2HdTO8WCKpFn5N9gDknatWFZMbCOiWx9AX2AesBXQH3gY2K6769GbH6Se9nbO\n00OAp4DtgO8BJ+XyrwJn5untc5z757jPA/qU/T566gP4EvB/wI153nEtJq6XAEfm6X7AUMe2kLhu\nBcwHBub5q4HDHdua4/kRYBdgTkVZZ2LZ0pJiFjAlT/8KmFb2eyv70U5sP9ry+QPOdGyLiWsuHwvc\nDCwAhjuuhX1m/x64Feif50c4toXF9k7g43l6OnBH0bEt44rPFGBeRCyMiJXAVcD+JdSj14qIFyPi\n4Tz9BvAksAWwH+nHJfn5gDy9P3BlRKyMiIWkD8yUbq10LyFpDLAP8FNaO+1wXOuUz+J+JCIuAoiI\ntyPiNRzbIvwVWAkMltQPGAy8gGNbk4i4G3ilTXFnYrmbpNHARhExK693acU2661qsY2IWyNidZ69\nHxiTpx3bDmrnMwtwFnBSmzLHtRPaie2xwHfyb1gi4qVc7th2QjuxXUw6KQowDHg+TxcW2zISny2A\n5yrmF+Uyq4GkrUgZ8/3AyIhYkhctAUbm6c1JcW7hmLfvh8BXgNUVZY5r/cYDL0m6WNIfJF2g1Ouj\nY1uniFgK/AB4lpTwvBoRt+LYFqmzsWxb/jyOcUccSTpjC45tXSTtDyyKiEfbLHJc67c1sKek+yTd\nKWlyLnds6/c14AeSngVmACfn8sJiW0bi494UCiJpCHAdcHxEvF65LNI1v7XF2n+HNiR9AvhzRMzm\n3V20v8NxrVk/YCJwXkRMBP5GOsC9w7GtjaQJwAmky/+bA0OUBpJ+h2NbnA7E0mog6evAWxFxRdl1\n6e0kDQZOAZori0uqTiPqB2wSER8inSi9puT6NJILgeMiYhzwReCiol+gjMTneVK70xZjeXe2Zh0g\nqT8p6bksIm7IxUskjcrLRwN/zuVtYz6G1suH1moPYD9JC4Argb0lXYbjWoRFpLOPD+T5a0mJ0IuO\nbd0mkwaNfjki3gauB3bHsS1SZ44Bi3L5mDbljnE7JH2O1MT4sIpix7Z2E0gnQh7J32djgIckjcRx\nLcIi0nGW/J22WtJ7cGyLMCUifp6nr6W1GXZhsS0j8XkQ2FrSVpIGAAcDN5ZQj15LkkhZ8RMR8aOK\nRTeSbmomP99QUX6IpAGSxpMu087C3iUiTomIsRExHjgE+E1EfAbHtW4R8SLwnKRtctFU4HFgJo5t\nveYCH5I0KB8bpgJP4NgWqVPHgPx5/6tSz4UCPlOxjVWQNI101nz/iFhescixrVFEzImIkRExPn+f\nLQIm5uaajmv9bgD2BsjfaQMi4i84tkWYJ2mvPL038HSeLi623dV7Q+WD1FPDU6Sbk04uow69+QF8\nmHQPysPA7PyYBgwHbssflFuAYRXbnJLjPZfcY4Yfa43xXrT26ua4FhPTnYAHgEdIZ8uGOraFxfYk\nUiI5h3TzfX/HtuZYXkm6V+ot0v2oR9QSS2BS/nvMA84u+331hEeV2B4JPAP8qeK77DzHtua4rmj5\nzLZZPp/cq5vjWn9s8/H1shyrh4Amx7au2FYeayeT7ll/GPg9sEvRsfUApmZmZmZm1vBKGcDUzMzM\nzMysOznxMTMzMzOzhufEx8zMzMzMGp4THzMzMzMza3hOfMzMzMzMrOE58TEzMzMzs4bnxMfMzMzM\nzBqeEx8zMzMzM2t4/w9C6hHSVJJjrgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10d8092d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(log10((En_array[0:1800]-En_array[-1])/(En_array[0]-En_array[-1])))\n",
    "plot(log10((En_array_overrelaxed[0:1800]-En_array[-1])/(En_array[0]-En_array[-1])))\n",
    "plot(log10((En_array_fista[0:1800]-En_array[-1])/(En_array[0]-En_array[-1])))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise__ \n",
    "\n",
    "Try different values of a. You can try a=3,10,30,50,100."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We can note that the accelerated forward-backward is not monotonic: the cost function E is not decreasing along with the iterations and some oscillations are present."
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "<script>\n",
    "  $(document).ready(function(){\n",
    "      $('div.prompt').hide();\n",
    "  });\n",
    "</script>"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.4.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}