{
 "metadata": {
  "language": "Julia",
  "name": "",
  "signature": "sha256:45171162088efc3437fbfcc626939b6f52664d7d7b5c6d07b04db4a3931b2156"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "using Convex\n",
      "using SCS\n",
      "\n",
      "# passing in verbose=0 to hide output from SCS\n",
      "solver = SCSSolver(verbose=0)\n",
      "set_default_solver(solver);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Linear program\n",
      "$$\n",
      "\\begin{array}{ll}\n",
      "  \\mbox{maximize} & c^T x \\\\\n",
      "  \\mbox{subject to} & A x <= b\\\\\n",
      "  & x \\geq 1 \\\\\n",
      "  & x \\leq 10 \\\\\n",
      "  & x_2 \\leq 5 \\\\\n",
      "  & x_1 + x_4 - x_2 \\leq 10 \\\\\n",
      "\\end{array}\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = Variable(4)\n",
      "c = [1; 2; 3; 4]\n",
      "A = eye(4)\n",
      "b = [10; 10; 10; 10]\n",
      "p = minimize(dot(c, x)) # or c' * x\n",
      "p.constraints += A * x <= b\n",
      "p.constraints += [x >= 1; x <= 10; x[2] <= 5; x[1] + x[4] - x[2] <= 10]\n",
      "solve!(p)\n",
      "\n",
      "println(round(p.optval, 2))\n",
      "println(round(x.value, 2))\n",
      "println(evaluate(x[1] + x[4] - x[2]))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "10"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        ".0\n",
        "["
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "1.0\n",
        " 1.0\n",
        " 1.0\n",
        " 1.0]\n",
        "[0.9999794207077799]\n"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Matrix Variables and promotions\n",
      "$$\n",
      "\\begin{array}{ll}\n",
      "  \\mbox{minimize} & \\| X \\|_F + y \\\\\n",
      "  \\mbox{subject to} & 2 X <= 1\\\\\n",
      "  & X' + y >= 1 \\\\\n",
      "  & X >= 0 \\\\\n",
      "  & y >= 0 \\\\\n",
      "\\end{array}\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X = Variable(2, 2)\n",
      "y = Variable()\n",
      "# X is a 2 x 2 variable, and y is scalar. X' + y promotes y to a 2 x 2 variable before adding them\n",
      "p = minimize(sum(X) + y, 2 * X <= 1, X' + y >= 1, X >= 0, y >= 0)\n",
      "solve!(p)\n",
      "println(round(X.value, 2))\n",
      "println(y.value)\n",
      "p.optval"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[0"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        ".0 0.0\n",
        " 0.0 0.0]\n",
        "1.0000000807021356\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 3,
       "text": [
        "1.000001111507024"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Norm, exponential and geometric mean\n",
      "$$\n",
      "\\begin{array}{ll}\n",
      "  \\mbox{satisfy} & \\| x \\|_2 \\leq 100 \\\\\n",
      "  & e^{x_1} \\leq 5 \\\\\n",
      "  & x_2 \\geq 7 \\\\\n",
      "  & \\sqrt{x_3 x_4} \\geq x_2\n",
      "\\end{array}\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = Variable(4)\n",
      "p = satisfy(norm(x) <= 100, exp(x[1]) <= 5, x[2] >= 7, geo_mean(x[3], x[4]) >= x[2])\n",
      "solve!(p, SCSSolver(verbose=0))\n",
      "println(p.status)\n",
      "x.value"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Optimal\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 4,
       "text": [
        "4x1 Array{Float64,2}:\n",
        "  0.0    \n",
        "  8.65837\n",
        " 14.6655 \n",
        " 14.6655 "
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### SDP cone and Eigenvalues\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y = Semidefinite(2)\n",
      "p = maximize(lambda_min(y), trace(y)<=6)\n",
      "solve!(p, SCSSolver(verbose=0))\n",
      "p.optval"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": [
        "3.0000029948079856"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = Variable()\n",
      "y = Variable((2, 2))\n",
      "# SDP constraints\n",
      "p = minimize(x + y[1, 1], isposdef(y), x >= 1, y[2, 1] == 1)\n",
      "solve!(p)\n",
      "y.value"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "2x2 Array{Float64,2}:\n",
        " 0.00352332    1.00018\n",
        " 1.00018     286.415  "
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Mixed integer program\n",
      "$$\n",
      "\\begin{array}{ll}\n",
      "  \\mbox{minimize} & sum(x) \\\\\n",
      "    \\mbox{subject to} & x \\in \\mathbb{Z} \\\\\n",
      "  & x >= 0.5 \\\\\n",
      "\\end{array}\n",
      "$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "using GLPKMathProgInterface\n",
      "x = Variable(4, :Int)\n",
      "p = minimize(sum(x), x >= 0.5)\n",
      "solve!(p, GLPKSolverMIP())\n",
      "x.value"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "4x1 Array{Float64,2}:\n",
        " 1.0\n",
        " 1.0\n",
        " 1.0\n",
        " 1.0"
       ]
      }
     ],
     "prompt_number": 7
    }
   ],
   "metadata": {}
  }
 ]
}