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     "source": [
      "# Entropy Maximization\n",
      "\n",
      "Here is a constrained entropy maximization problem:\n",
      "\n",
      "\\begin{array}{ll}\n",
      "    \\mbox{maximize}   & -\\sum_{i=1}^n x_i \\log x_i \\\\\n",
      "    \\mbox{subject to} & \\mathbf{1}' x = 1 \\\\\n",
      "                  & Ax \\leq b\n",
      "\\end{array}\n",
      "\n",
      "where $x \\in \\mathbf{R}^n$ is our optimization variable and $A \\in \\mathbf{R}^{m \\times n}, b \\in \\mathbf{R}^{m}$.\n",
      "\n",
      "To solve this, we can simply use the `entropy` operation Convex.jl provides."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "using Convex, SCS\n",
      "\n",
      "n = 25;\n",
      "m = 15;\n",
      "A = randn(m, n); \n",
      "b = rand(m, 1); \n",
      "\n",
      "x = Variable(n);\n",
      "problem = maximize(entropy(x), sum(x) == 1, A * x <= b)\n",
      "solve!(problem, SCSSolver(verbose=0))\n",
      "\n",
      "println(problem.optval)\n",
      "println(x.value)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "3."
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