line-search.ipynb

{
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  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots\n",
    "# Borrowing heavily from D. Bindel's notes:\n",
    "# https://github.com/dbindel/cs4220-s20/blob/master/lec/jupyter/2020-04-24.ipynb"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "H (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = x * atan(x) - log(1.0 + x^2) / 2.0\n",
    "g(x) = atan(x)\n",
    "H(x) = 1.0 / (1.0 + x^2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "xs = collect(range(-2, 2, length=200));\n",
    "fx = [f(x) for x in xs]\n",
    "plot(xs, fx, legend=false)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Warning: Skipped marker arg o.\n",
      "└ @ Plots /Users/arb/.julia/packages/Plots/KN96c/src/args.jl:862\n"
     ]
    },
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       "  236.501,87.9763 246.825,109.373 257.15,130.691 267.475,151.928 277.799,173.083 288.124,194.156 298.449,215.144 308.773,236.046 319.098,256.86 329.423,277.586 \n",
       "  339.748,298.222 350.072,318.766 360.397,339.216 370.722,359.571 381.046,379.83 391.371,399.99 401.696,420.05 412.021,440.008 422.345,459.862 432.67,479.61 \n",
       "  442.995,499.251 453.319,518.783 463.644,538.202 473.969,557.509 484.294,576.699 494.618,595.772 504.943,614.725 515.268,633.555 525.592,652.261 535.917,670.84 \n",
       "  546.242,689.29 556.566,707.608 566.891,725.791 577.216,743.837 587.541,761.744 597.865,779.508 608.19,797.127 618.515,814.598 628.839,831.918 639.164,849.085 \n",
       "  649.489,866.094 659.814,882.943 670.138,899.629 680.463,916.148 690.788,932.497 701.112,948.672 711.437,964.671 721.762,980.49 732.087,996.124 742.411,1011.57 \n",
       "  752.736,1026.82 763.061,1041.88 773.385,1056.74 783.71,1071.4 794.035,1085.85 804.359,1100.08 814.684,1114.1 825.009,1127.9 835.334,1141.48 845.658,1154.82 \n",
       "  855.983,1167.93 866.308,1180.81 876.632,1193.44 886.957,1205.82 897.282,1217.94 907.607,1229.81 917.931,1241.42 928.256,1252.76 938.581,1263.83 948.905,1274.62 \n",
       "  959.23,1285.12 969.555,1295.34 979.879,1305.26 990.204,1314.89 1000.53,1324.21 1010.85,1333.23 1021.18,1341.93 1031.5,1350.32 1041.83,1358.38 1052.15,1366.11 \n",
       "  1062.48,1373.51 1072.8,1380.57 1083.13,1387.3 1093.45,1393.67 1103.78,1399.69 1114.1,1405.36 1124.43,1410.68 1134.75,1415.62 1145.07,1420.21 1155.4,1424.42 \n",
       "  1165.72,1428.26 1176.05,1431.72 1186.37,1434.81 1196.7,1437.52 1207.02,1439.84 1217.35,1441.78 1227.67,1443.33 1238,1444.5 1248.32,1445.28 1258.65,1445.67 \n",
       "  1268.97,1445.67 1279.3,1445.28 1289.62,1444.5 1299.95,1443.33 1310.27,1441.78 1320.59,1439.84 1330.92,1437.52 1341.24,1434.81 1351.57,1431.72 1361.89,1428.26 \n",
       "  1372.22,1424.42 1382.54,1420.21 1392.87,1415.62 1403.19,1410.68 1413.52,1405.36 1423.84,1399.69 1434.17,1393.67 1444.49,1387.3 1454.82,1380.57 1465.14,1373.51 \n",
       "  1475.47,1366.11 1485.79,1358.38 1496.11,1350.32 1506.44,1341.93 1516.76,1333.23 1527.09,1324.21 1537.41,1314.89 1547.74,1305.26 1558.06,1295.34 1568.39,1285.12 \n",
       "  1578.71,1274.62 1589.04,1263.83 1599.36,1252.76 1609.69,1241.42 1620.01,1229.81 1630.34,1217.94 1640.66,1205.82 1650.99,1193.44 1661.31,1180.81 1671.63,1167.93 \n",
       "  1681.96,1154.82 1692.28,1141.48 1702.61,1127.9 1712.93,1114.1 1723.26,1100.08 1733.58,1085.85 1743.91,1071.4 1754.23,1056.74 1764.56,1041.88 1774.88,1026.82 \n",
       "  1785.21,1011.57 1795.53,996.124 1805.86,980.49 1816.18,964.671 1826.51,948.672 1836.83,932.497 1847.15,916.148 1857.48,899.629 1867.8,882.943 1878.13,866.094 \n",
       "  1888.45,849.085 1898.78,831.918 1909.1,814.598 1919.43,797.127 1929.75,779.508 1940.08,761.744 1950.4,743.837 1960.73,725.791 1971.05,707.608 1981.38,689.29 \n",
       "  1991.7,670.84 2002.03,652.261 2012.35,633.555 2022.67,614.725 2033,595.772 2043.32,576.699 2053.65,557.509 2063.97,538.202 2074.3,518.783 2084.62,499.251 \n",
       "  2094.95,479.61 2105.27,459.862 2115.6,440.008 2125.92,420.05 2136.25,399.99 2146.57,379.83 2156.9,359.571 2167.22,339.216 2177.55,318.766 2187.87,298.222 \n",
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   "source": [
    "newton_xs = [1.3]\n",
    "for k = 1:20\n",
    "    xk = newton_xs[end]\n",
    "    pk = - g(xk) / H(xk)\n",
    "    push!(newton_xs, xk + pk)\n",
    "end\n",
    "\n",
    "plot(xs, fx, legend=false)\n",
    "plot!(newton_xs, [f(x) for x in newton_xs], m=\"o\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Warning: Skipped marker arg o.\n",
      "└ @ Plots /Users/arb/.julia/packages/Plots/KN96c/src/args.jl:862\n"
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     "metadata": {},
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   ],
   "source": [
    "newton_xs = [1.5]\n",
    "for k = 1:3\n",
    "    xk = newton_xs[end]\n",
    "    pk = -g(xk) / H(xk)\n",
    "    push!(newton_xs, xk + pk)\n",
    "end\n",
    "\n",
    "plot(xs, fx, legend=false)\n",
    "plot!(newton_xs, [f(x) for x in newton_xs], m=\"o\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "find_step_size (generic function with 1 method)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function find_step_size(f, x, p)\n",
    "   fx = f(x) \n",
    "    α = 1.0\n",
    "    while f(x + α * p) > fx\n",
    "        α /= 2.0\n",
    "    end\n",
    "    return α\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Warning: Skipped marker arg o.\n",
      "└ @ Plots /Users/arb/.julia/packages/Plots/KN96c/src/args.jl:862\n"
     ]
    },
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       "  959.23,1285.12 969.555,1295.34 979.879,1305.26 990.204,1314.89 1000.53,1324.21 1010.85,1333.23 1021.18,1341.93 1031.5,1350.32 1041.83,1358.38 1052.15,1366.11 \n",
       "  1062.48,1373.51 1072.8,1380.57 1083.13,1387.3 1093.45,1393.67 1103.78,1399.69 1114.1,1405.36 1124.43,1410.68 1134.75,1415.62 1145.07,1420.21 1155.4,1424.42 \n",
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       "  1578.71,1274.62 1589.04,1263.83 1599.36,1252.76 1609.69,1241.42 1620.01,1229.81 1630.34,1217.94 1640.66,1205.82 1650.99,1193.44 1661.31,1180.81 1671.63,1167.93 \n",
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       "  1888.45,849.085 1898.78,831.918 1909.1,814.598 1919.43,797.127 1929.75,779.508 1940.08,761.744 1950.4,743.837 1960.73,725.791 1971.05,707.608 1981.38,689.29 \n",
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       "  2198.2,277.586 2208.52,256.86 2218.84,236.046 2229.17,215.144 2239.49,194.156 2249.82,173.083 2260.14,151.928 2270.47,130.691 2280.79,109.373 2291.12,87.9763 \n",
       "  \n",
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       "  \"/>\n",
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       "</svg>\n"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "newton_xs = [1.5]\n",
    "for k = 1:5\n",
    "    xk = newton_xs[end]\n",
    "    pk = -g(xk) / H(xk)\n",
    "    αk = find_step_size(f, xk, pk)\n",
    "    push!(newton_xs, xk + αk * pk)\n",
    "end\n",
    "\n",
    "plot(xs, fx, legend=false)\n",
    "plot!(newton_xs, [f(x) for x in newton_xs], m=\"o\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "H2 (generic function with 1 method)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f2(x) = 19.0 * x^2 - 4.0 * x^4 + (7.0 / 9.0) * x^6\n",
    "g2(x) = 38.0 * x - 16 * x^3 + (14.0 / 3.0) *x^5\n",
    "H2(x) = 38.0 - 48.0 *x^2 + (70.0 / 3.0) * x^4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
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     "execution_count": 9,
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   "source": [
    "xs = collect(range(-1.2, 1.2, length=100));\n",
    "fx = [f2(x) for x in xs]\n",
    "plot(xs, fx, legend=false)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Warning: Skipped marker arg o.\n",
      "└ @ Plots /Users/arb/.julia/packages/Plots/KN96c/src/args.jl:862\n"
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     "execution_count": 10,
     "metadata": {},
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   ],
   "source": [
    "newton_xs = [1.01]\n",
    "for k = 1:20\n",
    "    xk = newton_xs[end]\n",
    "    pk = -g2(xk) / H2(xk)\n",
    "    αk = find_step_size(f2, xk, pk)\n",
    "    push!(newton_xs, xk + αk * pk)\n",
    "end\n",
    "\n",
    "plot(xs, fx, legend=false)\n",
    "plot!(newton_xs, [f2(x) for x in newton_xs], m=\"o\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "21-element Array{Float64,1}:\n",
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   ],
   "source": [
    "newton_xs"
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  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "find_step_size_armijo (generic function with 1 method)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function find_step_size_armijo(f, x, p, g, c1)\n",
    "    fx = f(x) \n",
    "    α = 1.0\n",
    "    factor = 0.9\n",
    "    while f(x + α * p) > fx || f(x + α * p) > fx + c1 * α * g' * p\n",
    "        α *= factor\n",
    "    end\n",
    "    return α\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Warning: Skipped marker arg o.\n",
      "└ @ Plots /Users/arb/.julia/packages/Plots/KN96c/src/args.jl:862\n"
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      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "newton_xs = [1.01]\n",
    "for k = 1:20\n",
    "    xk = newton_xs[end]\n",
    "    pk = -g2(xk) / H2(xk)\n",
    "    αk = find_step_size_armijo(f2, xk, pk, g2(xk), 1e-3)\n",
    "    push!(newton_xs, xk + αk * pk)\n",
    "end\n",
    "\n",
    "plot(xs, fx, legend=false)\n",
    "plot!(newton_xs, [f2(x) for x in newton_xs], m=\"o\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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