![](\"imgs/8/cal7-18.png\")
"
]
},
{
"cell_type": "code",
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{
"data": {
"text/plain": [
"![](\"imgs/8/cal7-20.png\")
\n",
"\n",
"Since the domain is in the rectangular form, there are only three\n",
" possibilities in each direction:\n",
"1. $\\mathbf{x}^{\\ast}$ is an interior point of domain, then $f_i(\\mathbf{x}^{\\ast}) = 0$ for any $i$ since it is smooth.\n",
"- It is at boundary and suppose that it is increasing, then $(x^i)^{\\ast} = b^i$ and $f_i (\\mathbf{x}^{\\ast}) > 0$.\n",
"- It is at boundary and suppose that it is decreasing, then $(x^i)^{\\ast} = a^i_{}$ and $f_i (\\mathbf{x}^{\\ast}) < 0$.\n",
"\n",
"And these prove this theorem. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Similarly, we have the following theorem to describe the behavior as minimum\n",
"of function within bounded region:\n",
"\n",
"Theorem\n",
"---\n",
" Suppose that\n",
"$$ \\text{Domain} (f) = \\left\\{ \\vec{x} | x_i \\in [a_i,\n",
" b_i], \\text{ for all $1 \\leqslant i \\leqslant n$:} \\} \\right.\n",
" $$\n",
" and $f (\\mathbf{x})$ is smooth with all its first order derivatives.\n",
" Suppose that $f (\\mathbf{x})$ attains it global minimum at\n",
" $\\mathbf{x}=\\mathbf{x}^{\\ast}$, then one or both following condition(s)\n",
" must hold:\n",
"1. $f_i (\\mathbf{x}^{\\ast}) \\geqslant 0$ and $((x^i)^{\\ast} - a^i_{})\n",
" f_i (\\mathbf{x}^{\\ast}) = 0$;\n",
"- $f_i (\\mathbf{x}^{\\ast}) \\leqslant 0$ and $(b^i - (x^i)^{\\ast})\n",
" f_i (\\mathbf{x}^{\\ast}) = 0$\n",
"\n",
" for all $1 \\leqslant i \\leqslant n$.\n",
"\n",
" \n",
"**Proof**\n",
"\n",
"![](\"imgs/8/cal7-21.png\")
\n",
"\n",
"There are only three possibilities in each direction:\n",
"1. $\\mathbf{x}^{\\ast}$ is an interior point of domain, then $f_i\n",
" (\\mathbf{x}^{\\ast}) = 0$ for any $i$ since it is smooth.\n",
"- It is at boundary and suppose that it is decreasing, then $(x^i)^{\\ast} = b^i$ and $f_i (\\mathbf{x}^{\\ast}) < 0$.\n",
"- It is at boundary and suppose that it is increasing, then $(x^i)^{\\ast} = a^i_{}$ and $f_i (\\mathbf{x}^{\\ast}) > 0$.\n",
"\n",
" And these prove this theorem.\n",
"\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example\n",
"---\n",
" Suppose that one factory inputs its goods from two different supply plants,\n",
" $A$ and $B$, with different costs, $4$ and $6$ each respective. And suppose\n",
" the price function in the market is the same and is decided as $p (x, y) =\n",
" 100 - x - y$ where $x$ and $y$ are the demand functions. Discuss the the\n",
" maxima problem due to the following situations:\n",
"1. $0 \\leqslant x, y$\n",
"- $0 \\leqslant x \\leqslant 30$, $0 \\leqslant y \\leqslant 30$\n",
"\n",
"**Solve**\n",
" \n",
"**1.** The profit function is\n",
"\n",
"\\begin{eqnarray*}\n",
" P (x, y) & = & p (x, y) (x + y) - 4 x - 6 y\\\\\n",
" & = & 100 (x + y) - (x + y)^2 - 4 x - 6 y\n",
" \\end{eqnarray*}\n",
" with all its first derivative:\n",
"\n",
"\\begin{eqnarray*}\n",
" P_x (x, y) & = & 100 - 2 (x + y) - 4\\\\\n",
" P_y (x, y) & = & 100 - 2 (x + y) - 6\n",
" \\end{eqnarray*}\n",
" Obviously, it is impossible to get the equations simultaneously:\n",
"\n",
"\\begin{eqnarray*}\n",
" P_x (x, y) = 0 & \\Longrightarrow & 100 - 2 (x + y) - 4 = 0\\\\\n",
" P_y (x, y) = 0 & \\Longrightarrow & 100 - 2 (x + y) - 6 = 0\n",
" \\end{eqnarray*}\n",
"\n",
"since it is absolutely inconsistent. Does it imply no maximum for $P (x,y)$? Certainly not. Think about it: if you can input more cheaper goods,\n",
"$4$ each, why not do so? The first derivative test can only be used to\n",
"confirm that the maximum does not occur at the point, i.e. satisfying $P_x= P_y = 0$.\n",
" \n",
"\n",
"In this case, we always choose to input from $A$ with cost $4$ each but not from $B$. This just implies:\n",
"\n",
"\\begin{eqnarray*}\n",
" y = 0 & \\text{ and } & P (x, y) = 100 x - x^2 - 4 x\n",
" \\end{eqnarray*}\n",
"\n",
"Then $P (x, y)$ attains its maximum at $(x, y) = (48, 0)$. Note that we have\n",
"\n",
"\\begin{eqnarray*}\n",
" \\vec{x}^{\\ast} = (x^{\\ast}, y^{\\ast}) = (48, 0) & \\Longrightarrow &\n",
" (x^{\\ast} - 0) P_x (\\vec{x}^{\\ast}) = 0 \\text{ and } P (\\vec{x}^{\\ast}) =\n",
" 0\\\\\n",
" & \\Longrightarrow & P_y (x^{\\ast}, y^{\\ast}) = - 2 - 2 y^{\\ast} < 0\n",
" \\text{ and } (y^{\\ast} - 0) P_x (\\vec{x}^{\\ast}) = 0\n",
" \\end{eqnarray*}\n",
"\n",
"**2.** Since the it is beneficial for us to input from $A$, certainly $x$ is equal to $30$, the maximal output from $A$. This concludes\n",
"\n",
"$$ P (x, y) = P (30, y) = 100 (30 + y) - (30 + y)^2 - 10 - 6 y $$\n",
"and\n",
"\n",
"\\begin{eqnarray*}\n",
" P_x (x, y) = 0 & \\Longrightarrow & 100 - 2 (x + y) - 4 = 0\\\\\n",
" P_y (x, y) = 0 & \\Longrightarrow & 100 - 2 (x + y) - 6 = 0\n",
" \\end{eqnarray*}\n",
" \n",
" \n",
"By the first derivative test, $P (x, y)$ will attains its maxima at $x =30$ (boundary point) and $y = 17$, coming from the fact, $P_y (30, y) =0$. In this case,\n",
"\n",
"\\begin{eqnarray*}\n",
" \\vec{x}^{\\ast} = (x^{\\ast}, y^{\\ast}) = (30, 17) & \\Longrightarrow & (30\n",
" - x^{\\ast}) P_x (\\vec{x}^{\\ast}) = 0 \\text{ and } P_x (\\vec{x}^{\\ast}) <\n",
" 0\\\\\n",
" & \\Longrightarrow & P_y (\\vec{x}^{\\ast}) = 0\n",
" \\end{eqnarray*}\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example\n",
"---\n",
"Two kinds of eggs, white and brown, are sold. The daily sales for white eggs\n",
" will be $W (x, y) = 30 - 15 x + 3 y$ and daily sales for brown eggs will be\n",
" $B (x, y) = 20 - 12 y + 2 x$ where $x, y$ are the sale prices for white and\n",
" brown eggs respectively. Then the revenue is\n",
"\n",
"\\begin{eqnarray*}\n",
" R (x, y) & = & x W (x, y) + y B (x, y)\\\\\n",
" & = & x (30 - 15 x + 3 y) + y (20 - 12 y + 2 x)\\\\\n",
" & = & 30 x + 20 y - 15 x^2 + 5 x y - 12 y^2\n",
" \\end{eqnarray*}\n",
" The critical point is calculated as follows:\n",
"\n",
"\\begin{eqnarray*}\n",
" \\nabla R = \\vec{0} & \\Longrightarrow & (30 - 30 x + 5 y, 20 + 5 x - 24 y)\n",
" = (0, 0)\\\\\n",
" & \\Longrightarrow & (x, y) = \\left( \\frac{164}{139}, \\frac{150}{139}\n",
" \\right)\n",
" \\end{eqnarray*}\n",
" Also the Hessian matrix is as follows:\n",
"\n",
"\\begin{eqnarray*}\n",
" H & = & \\left(\\begin{array}{cc}\n",
" - 30 & 5\\\\\n",
" 5 & - 24\n",
" \\end{array}\\right)\n",
" \\end{eqnarray*}\n",
" $R_1 = - 30 < 0$ and $|H| > 0$. Then at $(x, y) = \\left(\\frac{164}{139},\n",
" \\frac{150}{139}\\right)$, $R (x, y)$ attains its relative maximum and maximum too."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example\n",
"---\n",
"\n",
"Suppose that a firm produces one kind of output, $y$, by two inputs, $K$,\n",
" called capital, and $L$, called labor. Each unit of capital costs $u$ and\n",
" each unit of labor costs $v$. And the production function follows the\n",
" Cobb-Douglas relation:\n",
"\n",
"$$ P (K, L) = A K^{\\alpha} L^{\\beta} \\text{ where } A, \\alpha \\text{ and } \\beta\n",
" > 0 $$\n",
"If the price function is constant $p$, find the condition at which the\n",
" profit function attains its extremum and determine the condition at which\n",
" profit function actually attains its maximum at the critical point.\n",
" \n",
"**Sol**\n",
" \n",
"\n",
" The profit function, by definition, is\n",
"\n",
"\\begin{eqnarray*}\n",
" P (K, L) & = & R (K, L) - C (K, L)\\\\\n",
" & = & p A K^{\\alpha} L^{\\beta} - u K - v L\n",
" \\end{eqnarray*}\n",
"By the first order condition, the extremum occurs at which\n",
"\n",
"\\begin{eqnarray*}\n",
" 0 = \\frac{\\partial P}{\\partial K} & = p \\alpha A K^{\\alpha - 1}\n",
" L^{\\beta} - u & \\\\\n",
" 0 = \\frac{\\partial P}{\\partial L} & = p \\beta A K^{\\alpha} L^{\\beta - 1}\n",
" - v & \n",
" \\end{eqnarray*}\n",
"Then\n",
"\n",
"\\begin{eqnarray*}\n",
" & \\frac{L}{K} & = \\frac{u \\beta}{v \\alpha}\\\\\n",
" \\Longrightarrow & p \\alpha A K^{\\alpha - 1} \\left( \\frac{u \\beta}{v\\alpha} K \\right)^{\\beta} & = u\\\\\n",
" \\Longrightarrow & \\hat{K} & = \\left( \\frac{v^{\\beta} \\alpha^{\\beta -\n",
" 1}}{p A u^{\\beta - 1} \\beta^{\\beta}} \\right)^{\\frac{1}{\\alpha + \\beta -1}}\\\\\n",
" \\Longrightarrow & \\hat{L} & = \\left( \\frac{u^{\\alpha} \\beta^{\\alpha -\n",
" 1}}{p A v^{\\alpha - 1} \\alpha^{\\alpha}} \\right)^{\\frac{1}{\\alpha + \\beta - 1}}\n",
"\\end{eqnarray*}\n",
"\n",
"If $P (K, L)$ attains its maxima at $(\\hat{K}, \\hat{L})$, then\n",
"\n",
"\\begin{eqnarray*}\n",
" & |H| > 0 \\text{ and } |H_1 | < 0 & \\\\\n",
" \\Longrightarrow & \\frac{\\partial^2 P}{\\partial L^2} = p \\alpha (\\alpha -1) A K^{\\alpha - 2} L^{\\beta} < 0 & \\\\\n",
" & |H| = (p A)^2 \\alpha (\\alpha - 1) \\beta (\\beta - 1) K^{\\alpha + \\beta - 2} L^{\\alpha + \\beta - 2} & \\\\\n",
" & - (p A \\alpha \\beta)^2 K^{\\alpha + \\beta - 2} L^{\\alpha + \\beta - 2}< 0 & \\\\\n",
" & & \\\\\n",
" \\Longrightarrow & \\alpha < 1 \\text{ and } \\alpha + \\beta < 1 & \n",
"\\end{eqnarray*}"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def plot3d(x,y,z):\n",
" fig = plt.figure()\n",
" ax = Axes3D(fig)\n",
" ax.plot_surface(x, y, z, rstride=1, cstride=1, cmap=plt.cm.binary,alpha=0.9)\n",
" #ax.contour(x, y, z, lw=3, cmap=\"autumn_r\", linestyles=\"solid\", zdir='z',offset=0)\n",
" ax.set_xlabel('X')\n",
" ax.set_ylabel('Y')\n",
" ax.set_zlabel('Z')\n",
" ax.set_zlim(-7, 25)\n",
" #ax.scatter3D([1],[1],[0],color=(0,0,0));\n",
" ax.arrow(x=1,y=1,dx=0.1,dy=0.1)\n",
" t=np.linspace(0,2*np.pi,101)\n",
" xt=np.cos(t)\n",
" yt=np.sin(t)\n",
" zt=xt**2-xt*yt+yt**2-xt+yt-6\n",
" ax.plot3D(xt,yt,zt)\n",
" #ax.plot3D(xt,yt,np.sqrt(xt)+np.sqrt(yt))"
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {},
"outputs": [
{
"data": {
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NzU2eeuopOf8WiURYXV3FYDBw9epVTp06RbfbJRAIMDk5ic/no9VqMTc3RygUIpPJMDs7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\n8kjnhu7ubqxfv14hxfChkOLZBlEU8fTTT+Ohhx7CunXrkJSUh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"text/plain": [
"![](\"imgs/8/cal7-23.png\")
\n",
" \n",
"**Sol:** First we want to find the critical point(s) as follows:\n",
"\n",
"\\begin{eqnarray*}\n",
" \\vec{0} & = & (f_1, f_2)\\\\\n",
" & = & (2 x - y - 1, 2 y - x + 1)\\\\\n",
" \\Longrightarrow & & x = 1 / 3 \\text{ and } y = - 1 / 3\n",
" \\end{eqnarray*}\n",
"\n",
"**1.** Critical point is within this region and $f (1 / 3, - 1 / 3) = - 6\n",
" \\frac{1}{3}$. And the possible positions for $f (x, y)$ attaining its\n",
" extrema is at the boundary, $\\partial \\Omega = \\{(x, y) | x^2 + y^2 =\n",
" 1\\}$, i.e. $x = \\cos \\theta$ and $y = \\sin \\theta$, $0 \\le \\theta\n",
" \\le 2 \\pi$. On the boundary,\n",
" \n",
"\\begin{eqnarray*}\n",
" f (x, y) & = & f (\\cos \\theta, \\sin \\theta)\\\\\n",
" & = & \\cos^2 \\theta - \\sin \\theta \\cos \\theta + \\sin^2 \\theta - \\cos\n",
" \\theta + \\sin \\theta - 6\\\\\n",
" & = & - \\sin \\theta \\cos \\theta - \\cos \\theta + \\sin \\theta - 5\\\\\n",
" \\frac{d f}{d \\theta} = 0 & \\Longrightarrow & \\sin^2 \\theta - \\cos^2\n",
" \\theta + \\sin \\theta + \\cos \\theta = 0\\\\\n",
" & \\Longrightarrow & (\\sin \\theta + \\cos \\theta) (\\sin \\theta - \\cos\n",
" \\theta + 1) = 0\n",
" \\end{eqnarray*}\n",
" **a.** $\\sin \\theta + \\cos \\theta = 0$: $\\theta = 3 \\pi / 4$ and $7 \\pi /\n",
" 4$, this implies\n",
"\n",
"\\begin{eqnarray*}\n",
" f \\left( \\cos \\frac{3 \\pi}{4}, \\sin \\frac{3 \\pi}{4} \\right) & = & -\n",
" \\sin \\frac{3 \\pi}{4} \\cos \\frac{3 \\pi}{4} - \\cos \\frac{3 \\pi}{4} +\n",
" \\sin \\frac{3 \\pi}{4} - 5 = - 4 \\frac{1}{2}\\\\\n",
" f \\left( \\cos \\frac{7 \\pi}{4}, \\sin \\frac{7 \\pi}{4} \\right) & = & - 4\n",
" \\frac{1}{2}\n",
" \\end{eqnarray*}\n",
" **b.** $\\sin \\theta - \\cos \\theta + 1 = 0$: $\\theta = 0$ and $3 \\pi / 2$,\n",
" this implies\n",
"\n",
"\\begin{eqnarray*}\n",
" f (\\cos 0, \\sin 0) & = & - 6\\\\\n",
" f \\left(\\cos \\frac{3 \\pi}{2}, \\sin \\frac{3 \\pi}{2}\\right) & = & - 6\n",
" \\end{eqnarray*}\n",
"These conclude: maximum is $- 4 \\frac{1}{2}$ and minimum is $- 6\n",
" \\frac{1}{3}$ at $(x, y) = (1 / 3, - 1 / 3)$.\n",
"\n",
"**2.** In this square case, $\\partial \\Omega$ contains four components:\n", " **a.** $I_1 = \\{(x, y) | - 2 \\leqslant x \\leqslant 2, y = - 2\\}$;
\n", " **b.** $I_2 = \\{(x, y) | - 2 \\leqslant y \\leqslant 2, x = 2\\}$;
\n", " **c.** $I_3 = \\{(x, y) | - 2 \\leqslant x \\leqslant 2, y = 2\\}$;
\n", " **d.** $I_4 = \\{(x, y) | - 2 \\leqslant y \\leqslant 2, x = - 2\\}$;
\n", "\n", " - For $f (x, y) |_{I_1} = f (x, - 2) = x^2 + x - 4, - 2 \\leqslant x \\leqslant 2$:\n", " - maximum: $2$ at $(x, y) = (2, - 2)$;\n", " - minimum: $- 4 \\frac{1}{4}$ at $(x, y) = (- 1 / 2, - 2)$.\n", " - $f (x, y) |_{I_2} = f (2, y) = y^2 - y - 4^{}, - 2\\leqslant y \\leqslant 2$:\n", " - maximum: 2 at $(x, y) = (2, - 2)$;\n", " - minimum: $- 4 \\frac{1}{4}$ at $(x, y) = (2, 1 / 2)$.\n", " - $f (x, y) |_{I_3} = f (x, 2) = x^2 - 3 x, - 2 \\leqslant x\\leqslant 2$:\n", " - maximum: 10 at $(x, y) = (- 2, 2)$;\n", " - minimum: $- 9 / 4$ at $(x, y) = (3 / 2, 2)$.\n", " - $f (x, y) |_{I_4} = f (- 2, y) = y^2 + 3 y, - 2 \\leqslant y \\leqslant 2$:\n", " - maximum: 10 at $(x, y) = (- 2, 2)$;\n", " - minimum: $- 9 / 4$ at $(x, y) = (- 2, - 3 / 2)$.\n", "\n", "These conclude: maximum is $10$ at $(-2,2)$ and minimum is $- 6\n", " \\frac{1}{3}$ at $(x, y) = (1 / 3, - 1 / 3)$.\n", "\n", "**3.** the last case, we have these conclusions: maximum is $10$ at $(x, y) = (- 2,\n", " 2)$ and minimum is $- 6 \\frac{1}{3}$ at $(x, y) = (1 / 3, - 1 / 3)$.\n", "Since critical point is not included. Then the extrema might\n", " attain at the boundary $\\partial \\Omega$ as follows:\n", " - $I_1 = \\{(x, y) | 0 \\leqslant x \\leqslant 1, y = 0\\}$;\n", " - $I_2 = \\{(x, y) | 0 \\leqslant y \\leqslant 1, x = 1\\}$;\n", " - $I_3 = \\{(x, y) | 0 \\leqslant x \\leqslant 1, y = x\\}$;\n", "\n", "and\n", "\n", " - $f (x, y) |_{I_1} = f (x, 0) = x^2 - x - 6, 0 \\leqslant x\\leqslant 1$:\n", " - maximum : - 6 at $(x, y) = (0, 0)$ ;\n", " - minimum : $- 6 \\frac{1}{4}$ at $(x, y) = (1 / 2, 0)$.\n", " - $f (x, y) |_{I_2} = f (1, y) = y^2 - 6, 0 \\leqslant y\\leqslant 1$:\n", " - maximum: - 5 at $(x, y) = (1, 1)$;\n", " - minimum: - 6 at $(x, y) = (0, 1)$.\n", " - $f (x, y) |_{I_3} = f (x, x) = x^2 - 6, 0 \\leqslant x \\leqslant 1$:\n", " - maximum: - 5 at $(x, y) = (1, 1)$;\n", " - minimum: - 6 at }$(x, y) = (0, 0)$.\n", "\n", "These conclude:\n", " maximum is $- 5$ at $(x, y) = (1, 1)$ and minimum is $- 6 \\frac{1}{4}$ at\n", " $(x, y) = (1 / 2, 0)$ .\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Example\n", "---\n", "Find the extrema of $f (x, y, z) = (x - 1)^2 + (y - 1)^2 + (z - 1)^2$ for $x+ y + z \\le 4$ and $0 \\leqslant x, y, z$.\n", " \n", " The critical point is $(1, 1, 1)$ (with $f (1, 1, 1) = 0$) since\n", " $$ \\nabla f = 2 (x - 1, y - 1, z - 1) = \\vec{0} $$\n", " and it is within the domain. Since $f (x, y, z) \\geqslant 0$, then $f (x, y,\n", " z)$ attain its minimum at $(1, 1, 1)$. Another positions at which $f (x, y,\n", " z)$ possibly attains its extrema (or maximum) are on the boundary:\n", "- $I_1 : \\{(x, y, 0) | x + y \\leqslant 4, x, y \\geqslant 0\\}$;\n", "- $I_2 : \\{(x, 0, z) | x + z \\leqslant 4, x, z \\geqslant 0\\}$;\n", "- $I_3 : \\{(0, y, z) | y + z \\leqslant 4, y, z \\geqslant 0\\}$;\n", "- $I_4 : \\{(x, y, z) | x + y + z = 4, x, y, z \\geqslant 0\\}$\n", "\n", "
![](\"imgs/8/cal7-24.png\")
\n",
" \n",
"By the symmetry, the maximum in $I_1, I_2, I_3$ are the same. Therefore only\n",
" $I_1$ and $I_4$ are necessarily to be considered:\n",
"1. On $I_1$: $g (x, y) = f (x, y, 0) = (x - 1)^2 + (y - 1)^2 + 1$\n",
" - critical point is $(1, 1)$ (with $g (1, 1) = 1$) since\n",
" $$ \\nabla g = 2 (x - 1, y - 1) = \\vec{0} $$\n",
" - boundary points:\n",
" - $0 \\leqslant x \\leqslant 4, y = 0$: $g (x, y) = (x - 1)^2 + 2$\n",
" implies that maximum is 11 at $x = 4$.\n",
" - $0 \\leqslant y \\leqslant 4, x = 0$: $g (x, y) = (y - 1)^2 + 2$\n",
" implies that maximum is 11 at $y = 4$.\n",
" - $x + y = 4, x, y \\geqslant 0$: $g (x, y) = 2 x^2 - 8 x + 11$, $0\n",
" \\leqslant x \\leqslant 4$ implies that maximum is 11 at $x = 0$ or 4.\n",
"- On $I_4$: $f (x, y, z) = (x - 1)^2 + (y - 1)^2 + (z - 1)^2$ with $x\n",
" + y + z = 4, x, y, z \\geqslant 0$. This can be solved the following\n",
" technique, Lagrange multiplier, introduced in the following section:\n",
"\n",
"Let\n",
"\n",
"\\begin{eqnarray*}\n",
" L (x, y, z, \\lambda) & = & f (x, y, z) + \\lambda (4 - x - y - z)\\\\\n",
" & = & (x - 1)^2 + (y - 1)^2 + (z - 1)^2 + \\lambda (4 - x - y - z)\n",
" \\end{eqnarray*}\n",
"and\n",
"\n",
"\\begin{eqnarray*}\n",
" \\nabla L = \\vec{0} & \\Longrightarrow & 2 (x - 1) = 2 (y - 1) = 2 (z -\n",
" 1) = \\lambda\n",
" \\end{eqnarray*}\n",
" By symmetry, $x = y = z$, then\n",
"\n",
"\\begin{eqnarray*}\n",
" x + y + z = 4 & \\Rightarrow & x = y = z = 4 / 3\n",
" \\end{eqnarray*}\n",
" This implies $f (x, y, z)$ attains its minimum $1 / 3$ (why?) at this point $(4 / 3, 4 / 3, 4 / 3)$, but not maximum.\n",
"\n",
"\n",
"Therefore the maximum is 11 and minimum is 0.\n",
" \n",
" "
]
},
{
"cell_type": "code",
"execution_count": 87,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{x: l/2 + 1, z: l/2 + 1, y: l/2 + 1}"
]
},
"execution_count": 87,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x,y,z,l=symbols('x y z l')\n",
"f=(x-1)**2+(y-1)**2+(z-1)**2\n",
"cond=4-x-y-z\n",
"F=f+l*cond\n",
"solve(grad(F,[x,y,z]),[x,y,z])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Exercise\n",
"---\n",
"Suppose that $f (x, y) = 3 x + 4 y$. Find the extrema of $f (x, y)$ in the\n",
" following regions $\\Omega$ respectively:\n",
"1. $\\Omega = \\{(x, y) | - 2 \\leqslant x, y \\leqslant 2\\}$;\n",
"- $\\Omega = \\{(x, y) | x^2 + y^2 \\leqslant 2^2 \\}$;\n",
"\n",
"![](\"imgs/8/cal7-25.png\")
\n",
"1.\n",
"\\begin{eqnarray*}\n",
" \\partial \\Omega & = & \\{y = - 2\\} \\cup \\{x = 2\\} \\cup \\{y = 2\\} \\cup \\{x =\n",
" - 2\\}\\\\\n",
" & = & I_1 \\cup I_2 \\cup I_3 \\cup I_4\n",
" \\end{eqnarray*}\n",
" And the extrema appear at the ends, i.e. the four corners of the region, ($-\n",
" 2, - 2), (2, - 2), (2, 2)$ and (-2,2). And the functions values at these\n",
" points are $- 14, - 2, 14$ and $2$. Therefore, the maximum is $14$ and\n",
" minimum is $- 14$\n",
"- Since $\\partial \\Omega = \\{x^2 + y^2 = 4\\}$, consider the Lagrangian\n",
" function:\n",
" $$ L (x, y, \\lambda) = f (x, y) + \\lambda (4 - x^2 - y^2) $$\n",
" Then\n",
"\n",
"\\begin{eqnarray*}\n",
" \\nabla L = \\vec{0} & \\Longrightarrow & (3 - 2 \\lambda x, 4 - 2 \\lambda y,\n",
" 4 - x^2 - y^2) = \\vec{0}\\\\\n",
" & \\Longrightarrow & x = \\frac{3}{2 \\lambda}, y = \\frac{2}{\\lambda}\\\\\n",
" & \\Longrightarrow & y = \\frac{4}{3} x\\\\\n",
" & \\Longrightarrow & (x, y) = \\left( \\pm \\frac{6}{5}, \\pm \\frac{8}{5}\n",
" \\right)\n",
" \\end{eqnarray*}\n",
" Then the maximum is $f (6 / 5, 8 / 5) = 10$ and minimum is $f (- 6 / 5, - 8\n",
" / 5) = - 10$.\n",
"\n",
"\n",
"Exercise\n",
"---\n",
" Suppose that $f (x, y) = x^2 + y^2$. Find the extrema of $f (x, y)$ in the\n",
" following regions $\\Omega$ respectively:\n",
"1. $\\Omega = \\{(x, y) | - 2 \\leqslant x, y \\leqslant 2\\}$;\n",
"- $\\Omega = \\{(x, y) | 3 x + 4 y \\leqslant 5, 0 \\leqslant x, y\\}$;\n",
"\n",
"**1.**\n",
"\n",
"\\begin{eqnarray*}\n",
" \\partial \\Omega & = & \\{y = - 2\\} \\cup \\{x = 2\\} \\cup \\{y = 2\\} \\cup \\{x =\n",
" - 2\\}\\\\\n",
" & = & I_1 \\cup I_2 \\cup I_3 \\cup I_4\n",
" \\end{eqnarray*}\n",
" And the extrema appear at the ends, i.e. the four corners of the region, ($-\n",
" 2, - 2), (2, - 2), (2, 2)$ and (-2,2). The functions values at these points\n",
" are all 8 and $0 \\leqslant x^2 + y^2$. Therefore, the maximum is $8$ and\n",
" minimum is $0$.\n",
"\n",
"**2.** Since\n",
"\n",
"\\begin{eqnarray*}\n",
" \\partial \\Omega & = & \\{y = 0, 0 \\leqslant x \\leqslant 5 / 3\\} \\cup \\{x =\n",
" 0, 0 \\leqslant y \\leqslant 5 / 4\\} \\cup \\{3 x + 4 y = 5\\}\\\\\n",
" & = & I_1 \\cup I_2 \\cup I_3\n",
" \\end{eqnarray*}\n",
" Consider the Lagrangian function:\n",
" $$ L (x, y, \\lambda) = f (x, y) + \\lambda (5 - 3 x - 4 y) $$\n",
" Then\n",
" 1. $(x, y) \\in I_1$: $f (x, y) = x^2$ and $0 \\leqslant x \\leqslant 5 /\n",
" 3$. Then maximum is $25 / 9$ and minimum is 0.\n",
" - $(x, y) \\in I_2$: $f (x, y) = y^2$ and $0 \\leqslant y \\leqslant 5 /\n",
" 4$. Then maximum is $25 / 16$ and minimum is 0.\n",
" - $(x, y) \\in I_3$\n",
" \n",
"\\begin{eqnarray*}\n",
" \\nabla L = \\vec{0} & \\Longrightarrow & (2 x - 3 \\lambda x, 2 y - 4 \\lambda\n",
" y, 5 - 3 x - 4 y) = \\vec{0}\\\\\n",
" & \\Longrightarrow & x = \\frac{2}{3 \\lambda}, y = \\frac{1}{2 \\lambda}\\\\\n",
" & \\Longrightarrow & y = \\frac{3}{4} x\\\\\n",
" & \\Longrightarrow & (x, y) = \\left( \\frac{5}{6}, \\frac{5}{8} \\right)\n",
" \\end{eqnarray*}\n",
" Then the value of $f (x, y)$ is $625 / 576$ and is the minimum on $I_3$\n",
" (why?). These conclude that minimum is 0 and maximum is 25/9.\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"P.1125\n",
"---\n",
"Classify the types of extrema or saddle point of $f(x,y)$:\n", "**14. ** $f(x,y)=xy((3-x-y)$" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "
\n", " a. $y=0$ implies $x=0$ or $3-x=0$, i.e. $(0,0),(3,0)$; ($H<0$)
\n", " b. $3-2x-y=0$ implies \n", " - $x=0$ and $y=3$, ($H<0$)\n", " - $3-x-2y=0$ implies $(x,y)=(1,1)$ ($H>0,a<0$)\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**16. ** $f(x,y)=4y/(x^2+y^2+1)$" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "
\n", " a. $(x,y)=(0,1)$: $H>0,a<0$\n", " b. $(x,y)=(0,-1)$: $H>0,a>0$\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**36. ** Find the absolute extrema of $f(x,y)=3x^2+2xy+y^2$ on the triangle with vertices $(-2,-1),(1,-1),(1,2)$." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "
![](\"imgs/8/cal7-27.png\")
\n",
"\n",
"In real world, Lagrangian is more realistic than other kinds of optimization\n",
"problem: finding extrema under resources limited.\n",
"\n",
"For general case, $\\mathbf{x}\\in \\mathbb{R}^n$, the extrema of $f(\\mathbf{x}$ with constraint $g(\\mathbf{x}=0$ at $\\mathbf{x}_0$ should satisfy the following:\n",
"$$ \\nabla f(\\mathbf{x}_0)+\\lambda\\nabla g(\\mathbf{x}_0)=\\mathbf{0}$$\n",
"where $\\lambda\\ne0$ and $\\nabla g(\\mathbf{x}_0)\\ne\\mathbf{0}$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example\n",
"---\n",
" Suppose that the Cobb-Douglas function is\n",
" $$ P (x, y) = 100 x^{1 / 4} y^{3 / 4} $$\n",
" where $x$ is the unit of labor and $y$ is the unit of capital. Each unit of\n",
" labor costs $200$ and each of capital costs 100. If the total of 800 worth\n",
" of labor and capital is to be used, Find the maximum of $P (x, y)$.\n",
" \n",
"First note:\n",
"\n",
"the critical point at which $P (x, y)$ attains it maximum is also the\n",
" critical point for $\\ln P (x, y)$ that attains its maximum.\n",
"\n",
" \n",
" Since the constraint is\n",
"\n",
"$$ 200 x + 100 y = 800 $$\n",
" consider the Lagrangian function:\n",
"\n",
"\\begin{eqnarray*}\n",
" L (x, y, \\lambda) & = & \\ln P (x, y) + \\lambda (800 - 200 x - 100 y)\\\\\n",
" & = & \\ln 100 + \\frac{1}{4} \\ln x + \\frac{3}{4} \\ln y + \\lambda (800 -\n",
" 200 x - 100 y)\n",
" \\end{eqnarray*}\n",
"\n",
"Note: The critical value of $P (x, y)$ is\n",
" also critical value of $\\ln P (x, y)$!}\n",
"\n",
"Now the critical point(s) is as follows:\n",
"\n",
"\\begin{eqnarray*}\n",
" \\vec{0} & = & \\nabla L (x, y, \\lambda)\\\\\n",
" & = & \\left( \\frac{\\partial L}{\\partial x}, \\frac{\\partial L}{\\partial\n",
" y}, \\frac{\\partial L}{\\partial \\lambda} \\right)\\\\\n",
" & = & \\left( \\frac{1}{4 x} - 200 \\lambda, \\frac{3}{4 y} - 100 \\lambda,\n",
" 800 - 200 x - 100 y \\right)\\\\\n",
" \\Longrightarrow & & x = \\frac{1}{800 \\lambda}, y = \\frac{3}{400 \\lambda} \n",
" (\\text{ i.e. } y = 6 x)\\\\\n",
" \\Longrightarrow & & x = 1 \\text{ and } y = 6\n",
" \\end{eqnarray*}\n",
"\n",
"$(x, y) = (1, 6)$ is the only one critical point. Since $0 \\leqslant x, y\\le 8$, $P (x, y)$ has to be a maximum in such closed region.\n",
" Therefore, $P (x, y)$ attains it maximum at $(1, 6)$.\n",
" \n",
" "
]
},
{
"cell_type": "code",
"execution_count": 112,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Function of 100 x**(1/4)y**(3/4) with constraint -200*x - 100*y + 800:\n",
"\n",
"critical point (x,y) is (1,6)\n"
]
}
],
"source": [
"x,y,z,l=symbols('x y z l')\n",
"expf=100\n",
"f=log(100)+log(x)/4+3*log(y)/4\n",
"cond=800-200*x-100*y\n",
"F=f+l*cond\n",
"cpts=solve(grad(F,[x,y,z]),[x,y,z])\n",
"l0=solve(cond.subs({x:cpts[x],y:cpts[y]}),l)\n",
"print('Function of %s with constraint %s:\\n' %('100 x**(1/4)y**(3/4)',cond))\n",
"print('critical point (x,y) is (%s,%s)' %(cpts[x].subs(l,l0[0]),cpts[y].subs(l,l0[0])))\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Implement Python Lagrange's Multiplier solver\n",
"---"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def lagrangian(func,X,conditions,solvable=True):\n",
" l,m=symbols(\"lambda mu\")\n",
" if len(X)==2 and len(conditions)==1:\n",
" L=func+l*conditions[0]\n",
" cpts=solve([diff(L,x),diff(L,y),conditions[0]],[x,y,l])\n",
" print(\"Function, %s, subject to %s=0\\n---\" %(func,conditions[0])) \n",
" elif len(X)==3 and len(conditions)==1: \n",
" L=func+l*conditions[0]\n",
" cpts=solve([diff(L,x),diff(L,y),diff(L,z),conditions[0]],[x,y,z,l]) \n",
" print(\"Function, %s, subject to %s=0\\n---\" %(func,conditions[0])) \n",
" else:\n",
" L=func+l*conditions[0]+m* conditions[1]\n",
" cpts=solve([diff(L,x),diff(L,y),diff(L,z),conditions[0],conditions[1]],[x,y,z,l,m]) \n",
" print(\"Function, %s, subject to %s=0 and %s=0\\n---\" %(func,conditions[0],conditions[1]))\n",
" i=1\n",
" vals=[]\n",
" if solvable:\n",
" for cpt in cpts: \n",
" if len(X)==2:\n",
" funcVal=func.subs({x:cpt[0],y:cpt[1]})\n",
" print(\"%d֯ ). f = %s = %s at critical value (x,y)=(%s,%s)\" %(i,func,funcVal,cpt[0],cpt[1]))\n",
" else:\n",
" funcVal=func.subs({x:cpt[0],y:cpt[1],z:cpt[2]})\n",
" print(\"%d֯ ). f = %s = %s at critical value (x,y,z)=(%s,%s,%s)\" %(i,func, funcVal,cpt[0],cpt[1],cpt[2]))\n",
" vals.append(funcVal)\n",
" i+=1\n",
" print(\"---\\n\") \n",
" print(\"Maximum on the boundary is %s\" %max(vals))\n",
" print(\"Minimum on the boundary is %s\" %min(vals)) \n",
" else:\n",
" print(cpts) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example\n",
"---\n",
" Suppose you wishes to allocate your available time of $16$ hours in this\n",
" week between quizzes, English and Calculus, held in the next week. What\n",
" would you do in such way to maximize your grade average?\n",
"\n",
"**Solution**\n",
" \n",
"Suppose that\n",
"1. $f (t_1) = 20 + 20 \\sqrt{t_1} $: the time will be spent for\n",
"English course with $t_1$ hour per week,\n",
"- $g (t_2) = 50 + 3 t$ : the time will be spent for Calculus course\n",
" with $t_2$ hour per week.\n",
"\n",
"Then the problem is turned to be:\n",
"\n",
"\\begin{eqnarray*}\n",
" \\text{Maximize } S (t_1, t_2) & = & \\frac{f (t_1) + g (t_2)}{2}\\\\\n",
" \\text{ subject to } & & t_1 + t_2 = 16\n",
" \\end{eqnarray*}\n",
"Consider the Lagrangian:\n",
" $$ L (t_1, t_2) = 20 + 20 \\sqrt{t_1} + 50 + 3 t_2 + \\lambda (16 - t_1 -\n",
" t_2) $$\n",
"The extremum occurs at the place that satisfies:\n",
"\n",
"\\begin{eqnarray*}\n",
" \\frac{\\partial L}{\\partial t_1} = 0, & \\frac{\\partial L}{\\partial t_2} =\n",
" 0, & \\frac{\\partial L}{\\partial \\lambda} = 0.\n",
" \\end{eqnarray*}\n",
"And these imply:\n",
"\n",
"\\begin{eqnarray*}\n",
" & \\frac{10}{\\sqrt{t_1}} - \\lambda = 0, & 3 - \\lambda = 0\\\\\n",
" \\Longrightarrow & t_1 = \\left( \\frac{10}{3} \\right)^2 & \\text{ and } t_2 =\n",
" 16 - \\left( \\frac{10}{3} \\right)^2\n",
" \\end{eqnarray*}\n",
"How do I know at which the maximum occurs? Since $S (t_1, t_2)$ is\n",
"continuous for both $t_1$ and $t_2$ are in bounded intervals, the function\n",
"obtains its maximum and minimum. Comparing with the function values at\n",
"boundary, $S (t_1, t_2)$ will obtains its maximum at $(t_1, t_2) = (1,15)$ with the constraint $t_1 + t_2 = 16$.\n",
"\n",
"Theorem (Heron's formulae)\n",
"---\n",
"Suppose that The lengths of sides of triangle are $x, y$\n",
"and $z$ respectively and $x + y + z = l$. Then the area of this triangle is\n",
" $\\sqrt{s (s - x) (s - y) (s - z)}$ where $s = l / 2$.\n",
"\n",
"\n",
"Example\n",
"---\n",
" Suppose that the perimeter is $24$. Find the dimension of this triangle such\n",
" that it owns maximum area.\n",
" \n",
" Suppose $x, y, z$ are the lengths of sides of this triangle and let $A$ be\n",
" its area. Then\n",
"\n",
"\\begin{eqnarray*}\n",
" & \\text{Maximize} & A = \\sqrt{12 (12 - x) (12 - y) (12 - z)} \\text{ with }\n",
" x + y + z = 24\\\\\n",
" \\Longrightarrow & \\text{Maximize } & 12 (12 - x) (12 - y) (12 - z)\n",
" \\text{ with } x + y + z = 24\\\\\n",
" \\Longrightarrow & \\text{Maximize } & L (x, y, z, \\lambda) = \\ln (12 (12 -\n",
" x) (12 - y) (12 - z)) + \\lambda (24 - x - y - z)\n",
" \\end{eqnarray*}\n",
"Since\n",
"$$ L (x, y, z, \\lambda) = \\ln 12 + \\ln (12 - x) + \\ln (12 - y) + \\ln (12 -\n",
" z) + \\lambda (24 - x - y - z) $$\n",
" then the critical value of $L (x, y, z, \\lambda)$ can be found by the\n",
" following steps:\n",
"\n",
"\\begin{eqnarray*}\n",
" \\vec{0} & = & \\nabla L (x, y, z, \\lambda)\\\\\n",
" & = & \\left( \\frac{\\partial L}{\\partial x}, \\frac{\\partial L}{\\partial\n",
" y}, \\frac{\\partial L}{\\partial z}, \\frac{\\partial L}{\\partial \\lambda}\n",
" \\right)\\\\\n",
" & = & \\left( - \\frac{1}{12 - x} - 1, - \\frac{1}{12 - y} - 1, -\n",
" \\frac{1}{12 - z} - 1, 24 - x - y - z \\right)\n",
" \\end{eqnarray*}\n",
" This implies that $x = y = z = 8$, i.e. this triangle is equilateral. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Exercise\n",
"---\n",
"Assume that $P (x, y) = 100 x^{1 / 4} y^{3 / 4}$. Then $\\frac{\\partial\n",
" P}{\\partial x} = \\frac{25 y^{\\frac{3}{4}}}{x^{\\frac{3}{4}}}$ and\n",
" $\\frac{\\partial P}{\\partial y} = \\frac{75\n",
" x^{\\frac{1}{4}}}{y^{\\frac{1}{4}}}$.\n",
"1. $\\frac{\\partial P}{\\partial x} (1, 6) = 25 \\sqrt[4]{6^3}$\n",
"- $\\frac{\\partial P}{\\partial y} (1, 6) = 75 / \\sqrt[4]{6^{}}$\n",
"- $\\frac{\\partial P}{\\partial x} (1, 6) / \\frac{\\partial P}{\\partial\n",
" y} (1, 6) = 25 \\sqrt[4]{6^3} / (75 / \\sqrt[4]{6^{}}) = 6 / 3 = 2$\n",
"- Let $F (x, y) = P (x, y) + \\lambda (80 - 20 x - 10 y)$. Then\n",
"\n",
"$$\\text{ } 0 = \\frac{\\partial F}{\\partial x} = \\frac{25\n",
" y^{\\frac{3}{4}}}{x^{\\frac{3}{4}}} - 20 \\lambda \\text{ and } 0 =\n",
" \\frac{\\partial F}{\\partial y} = \\frac{75\n",
" x^{\\frac{1}{4}}}{y^{\\frac{1}{4}}} - 10 \\lambda $$\n",
"These imply $y = 6 x. \\text{Therefore}$ $F (x, y)$ attains its maximum\n",
" under the constraint $20 x + 10 y = 80$ as $x = 10$ and $y = 60$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example\n",
"---\n",
"Find the extrema of $f(x,y)=x^2-2y$ subject to $x^2+y^2=9$.\n",
"\n",
"**1.** Let $L=f+\\lambda(9-x^2-y^2)=x^2-2y+\\lambda(9-x^2-y^2)$.\n", "**2.**\n", "\\begin{eqnarray}\n", " \\vec{0} & = & \\nabla L (x, y, \\lambda)\\\\\n", " & = & \\left( \\frac{\\partial L}{\\partial x}, \\frac{\\partial L}{\\partial\n", " y}, \\frac{\\partial L}{\\partial \\lambda}\n", " \\right)\\\\\n", " & = & \\left( 2x-2\\lambda x, -2 - 2\\lambda y , 9-x^2-y^2 \\right)\n", "\\end{eqnarray}\n", "**3.**\n", " - $x=0\\to 0^2+y^2=9\\to y=\\pm3$\n", " - $\\lambda=1\\to y=-1\\to x=\\pm2\\sqrt2$\n", " - $f(0,-3)=6,f(0,3)=-6 (\\min),f(\\pm2\\sqrt2,-1)=10 (\\max)$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Example\n", "---\n", "Find the extrema of $f(x,y)=2x^2+y^2-2y+1$ subject to $x^2+y^2\\le4$.\n", "\n", "**0.** critical point:\n", "$$\\nabla f=(0,0)\\to(x,y)=(0,1)$$\n", "and $f(0,1)=0$.\n", "\n", "**1.** Let $L=f+\\lambda(4-x^2-y^2)=2x^2+y^2-2y+1+\\lambda(4-x^2-y^2)$.
\n", "**2.**\n", "\\begin{eqnarray}\n", " \\vec{0} & = & \\nabla L (x, y, \\lambda)\\\\\n", " & = & \\left( \\frac{\\partial L}{\\partial x}, \\frac{\\partial L}{\\partial\n", " y}, \\frac{\\partial L}{\\partial \\lambda}\n", " \\right)\\\\\n", " & = & \\left( 4x-2\\lambda x, 2y-2 - 2\\lambda y , 4-x^2-y^2 \\right)\n", "\\end{eqnarray}\n", "**3.**\n", " - $x=0\\to 0^2+y^2=4\\to y=\\pm2$\n", " - $\\lambda=2\\to y=-1\\to x=\\pm\\sqrt3$\n", " - $f(0,1)=0$ (minimum), $f(0,-2)=9,f(0,2)=1,f(\\pm\\sqrt3,-1)=10$ (maximum)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Example (With two constraints)\n", "---\n", "Find the extrema of $f(x,y,z)=3x+2y+4z$ subject to $x-y+2z=1$ and $x^2+y^2=4$.\n", "\n", "**1.** Let $L=f+\\lambda(1-x+y-2z)+\\mu(4-x^2-y^2)=3x+2y+4z+\\lambda(1-x+y-2z)+\\mu(4-x^2-y^2)$.
\n", "**2.** Find critical point(s):\n", "\\begin{eqnarray}\n", " \\vec{0\\!} & = & \\nabla L (x, y, z,\\lambda,\\mu)\\\\\n", " & = & \\left( \\frac{\\partial L}{\\partial x}, \\frac{\\partial L}{\\partial y}, \n", " \\frac{\\partial L}{\\partial z}, \\frac{\\partial L}{\\partial \\lambda},\n", " \\frac{\\partial L}{\\partial\\mu}\n", " \\right)\\\\\n", " & = & \\left( 3-\\lambda -2\\mu x, 2+\\lambda-2\\mu y,4-2\\lambda , ...,...\\right)\n", "\\end{eqnarray}\n", "**3.**\n", " - $\\lambda=2\\to x=1/(2\\mu) \\text{ and }y=2/\\mu$\n", " - $x^2+y^2=4\\to \\mu=\\pm\\sqrt{17}/4\\to(x,y,z)=\\left(\\pm2/\\sqrt{17},\\pm8/\\sqrt{17},\\frac{1}{2}\\left(1\\pm\\frac{6}{\\sqrt{17}}\\right)\\right)$\n", " - Maximum $f\\left(2/\\sqrt{17},8/\\sqrt{17},\\frac{1}{2}\\left(1+\\frac{6}{\\sqrt{17}}\\right)\\right)=2(1+\\sqrt{17})$,\n", " - Minimum $f\\left(-2/\\sqrt{17},-8/\\sqrt{17},\\frac{1}{2}\\left(1-\\frac{6}{\\sqrt{17}}\\right)\\right)=2(1-\\sqrt{17})$.\n", " " ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Exercise\n", "---\n", "1. Find the extrema of $f (x, y) = x y^2$ with $2 x^2 + y^2 = 12$.\n", "- Find the extrema of $f (x, y, z) = 10 x + 2 y + 6 z$ with $x^2 + y^2+ z^2 = 35$." ] }, { "cell_type": "code", "execution_count": 117, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "8*sqrt(2)\n" ] } ], "source": [ "#from sympy import jacobian\n", "#grad = lambda func, vars : Matrix(1,1,[func]).jacobian(vars)\n", "f1=x*y*y\n", "cond=12-2*x*x-y*y\n", "L=f1+l*cond\n", "cpts=solve(grad(L,[x,y,l]),[x,y,l])\n", "print(max([p[0]*p[1]*p[1] for p in cpts]))" ] }, { "cell_type": "code", "execution_count": 186, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Function, x*y**2, subject to -2*x**2 - y**2 + 12=0\n", "---\n", "1֯ ). f = x*y**2 = -8*sqrt(2) at critical value (x,y)=(-sqrt(2),-2*sqrt(2))\n", "2֯ ). f = x*y**2 = -8*sqrt(2) at critical value (x,y)=(-sqrt(2),2*sqrt(2))\n", "3֯ ). f = x*y**2 = 8*sqrt(2) at critical value (x,y)=(sqrt(2),-2*sqrt(2))\n", "4֯ ). f = x*y**2 = 8*sqrt(2) at critical value (x,y)=(sqrt(2),2*sqrt(2))\n", "5֯ ). f = x*y**2 = 0 at critical value (x,y)=(-sqrt(6),0)\n", "6֯ ). f = x*y**2 = 0 at critical value (x,y)=(sqrt(6),0)\n", "---\n", "\n", "Maximum on the boundary is 8*sqrt(2)\n", "Minimum on the boundary is -8*sqrt(2)\n" ] } ], "source": [ "f=x*y*y\n", "cond=12-2*x*x-y*y\n", "lagrangian(f,[x,y],[cond])" ] }, { "cell_type": "code", "execution_count": 187, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Function, 10*x + 2*y + 6*z, subject to -x**2 - y**2 - z**2 + 35=0\n", "---\n", "1֯ ). f = 10*x + 2*y + 6*z = -70 at critical value (x,y,z)=(-5,-1,-3)\n", "2֯ ). f = 10*x + 2*y + 6*z = 70 at critical value (x,y,z)=(5,1,3)\n", "---\n", "\n", "Maximum on the boundary is 70\n", "Minimum on the boundary is -70\n" ] } ], "source": [ "f=10*x+2*y+6*z\n", "cond=35-x*x-y*y-z*z\n", "lagrangian(f,[x,y,z],[cond])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Exercise p1138\n", "---\n", "**10.** Find the extrema of $f(x,y)=x^2+y^2$ with $x^4+y^4=1$." ] }, { "cell_type": "code", "execution_count": 84, "metadata": { "collapsed": true }, "outputs": [], "source": [ "from sympy import symbols,solve\n", "x,y,z,l,m=symbols(\"x y z l m\",real=True)" ] }, { "cell_type": "code", "execution_count": 51, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Maximum is sqrt(2)\n", "Minimum is 1\n" ] } ], "source": [ "f1=x*x+y*y\n", "cond=1-x**4-y**4\n", "L=f1+l*cond\n", "cpts=solve(grad(L,[x,y,l]),[x,y,l])\n", "print(\"Maximum is %s\" %max([p[0]*p[0]+p[1]*p[1] for p in cpts]))\n", "print(\"Minimum is %s\" %min([p[0]*p[0]+p[1]*p[1] for p in cpts]))" ] }, { "cell_type": "code", "execution_count": 184, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Function, x**2 + y**2, subject to -x**4 - y**4 + 1=0\n", "---\n", "1֯ ). f = x**2 + y**2 = 1 at critical value (x,y)=(-1,0)\n", "2֯ ). f = x**2 + y**2 = 1 at critical value (x,y)=(0,-1)\n", "3֯ ). f = x**2 + y**2 = 1 at critical value (x,y)=(0,1)\n", "4֯ ). f = x**2 + y**2 = 1 at critical value (x,y)=(1,0)\n", "5֯ ). f = x**2 + y**2 = sqrt(2) at critical value (x,y)=(-2**(3/4)/2,-2**(3/4)/2)\n", "6֯ ). f = x**2 + y**2 = sqrt(2) at critical value (x,y)=(-2**(3/4)/2,2**(1/4)*sqrt(2)/2)\n", "7֯ ). f = x**2 + y**2 = sqrt(2) at critical value (x,y)=(2**(1/4)*sqrt(2)/2,-2**(3/4)/2)\n", "8֯ ). f = x**2 + y**2 = sqrt(2) at critical value (x,y)=(2**(1/4)*sqrt(2)/2,2**(1/4)*sqrt(2)/2)\n", "---\n", "\n", "Maximum on the boundary is sqrt(2)\n", "Minimum on the boundary is 1\n" ] } ], "source": [ "f1=x*x+y*y\n", "cond=1-x**4-y**4\n", "lagrangian(f1,[x,y],[cond])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "**14.** Find the extrema of $f(x,y)=x^2+y^2+z^2$ with $y-x=1$." ] }, { "cell_type": "code", "execution_count": 66, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "{l: 1, x: -1/2, y: 1/2, z: 0}" ] }, "execution_count": 66, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f2=x*x+y*y+z*z\n", "cond2=1+x-y\n", "L2=f2+l*cond2\n", "cpts=solve(grad(L2,[x,y,z,l]),[x,y,z,l])\n", "#print(\"Maximum is %s\" %max([p[0]*p[0]+p[1]*p[1]+p[2]*p[2] for p in cpts]))\n", "#print(\"Minimum is %s\" %min([p[0]*p[0]+p[1]*p[1]+p[2]*p[2] for p in cpts]))\n", "cpts" ] }, { "cell_type": "code", "execution_count": 183, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Function, x**2 + y**2 + z**2, subject to x - y + 1=0\n", "---\n", "{x: -1/2, y: 1/2, z: 0, lambda: 1}\n" ] } ], "source": [ "f2=x*x+y*y+z*z\n", "cond2=1+x-y\n", "\n", "lagrangian(f2,[x,y,z],[cond2],solvable=False)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since $x^2+y^2+z^2\\ge0$, only minimum attains and at $(x,y,z)=(-1/2,1/2,0)$, i.e. min=$1/2$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "**18.** Find the extrema of $f(x,y)=x+y+z$ with $x^2+y^2=1$ with $x+z=2$." ] }, { "cell_type": "code", "execution_count": 70, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Maximum is 3\n", "Minimum is 1\n" ] } ], "source": [ "f3=x+y+z\n", "cond31=1-x**2-y**2\n", "cond32=2-x-z\n", "L3=f3+l*cond31+m*cond32\n", "cpts=solve(grad(L3,[x,y,z,l,m]),[x,y,z,l,m])\n", "print(\"Maximum is %s\" %max([p[0]+p[1]+p[2] for p in cpts]))\n", "print(\"Minimum is %s\" %min([p[0]+p[1]+p[2] for p in cpts]))\n" ] }, { "cell_type": "code", "execution_count": 188, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Function, x + y + z, subject to -x**2 - y**2 + 1=0 and -x - z + 2=0\n", "---\n", "1֯ ). f = x + y + z = 1 at critical value (x,y,z)=(0,-1,2)\n", "2֯ ). f = x + y + z = 3 at critical value (x,y,z)=(0,1,2)\n", "---\n", "\n", "Maximum on the boundary is 3\n", "Minimum on the boundary is 1\n" ] } ], "source": [ "f3=x+y+z\n", "cond31=1-x**2-y**2\n", "cond32=2-x-z\n", "\n", "lagrangian(f3,[x,y,z],[cond31,cond32])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "**22.** Find the extrema of $f(x,y)=x^2y$ with $4x^2+y^2\\le4$." ] }, { "cell_type": "code", "execution_count": 140, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[(0, 0)]" ] }, "execution_count": 140, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f=x**2*y\n", "solve(grad(f,[x,y]),[x,y])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Actually, set of critical points is $(x,y)=\\{(0,y),y\\in(-2,2)\\}$, infinite points included." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Function, x**2*y, subject to -4*x**2 - y**2 + 4=0\n", "---\n", "1֯ ). f = x**2*y = 0 at critical value (x,y)=(0,-2)\n", "2֯ ). f = x**2*y = 0 at critical value (x,y)=(0,2)\n", "3֯ ). f = x**2*y = -4*sqrt(3)/9 at critical value (x,y)=(-sqrt(6)/3,-2*sqrt(3)/3)\n", "4֯ ). f = x**2*y = 4*sqrt(3)/9 at critical value (x,y)=(-sqrt(6)/3,2*sqrt(3)/3)\n", "5֯ ). f = x**2*y = -4*sqrt(3)/9 at critical value (x,y)=(sqrt(6)/3,-2*sqrt(3)/3)\n", "6֯ ). f = x**2*y = 4*sqrt(3)/9 at critical value (x,y)=(sqrt(6)/3,2*sqrt(3)/3)\n", "---\n", "\n", "Maximum on the boundary is 4*sqrt(3)/9\n", "Minimum on the boundary is -4*sqrt(3)/9\n" ] } ], "source": [ "f=x*x*y\n", "cond=4-4*x**2-y**2\n", "lagrangian(f,[x,y],[cond])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "After all, Maximum is $4\\sqrt3/9$, and Minimum is $-4\\sqrt3/9$.\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Method of Least Squares\n", "---\n", "\n", "Suppose that there are $n$ paired data, $(x_1, y_1), (x_2, y_2), \\cdots,\n", "(x_n, y_n)$, that were observed in one experiment. Can we find the approximate\n", "relation between $X = x_i$ and $Y = y_i$?\n", "\n", "\n", "
![](\"imgs/8/cal7-28.png\")
\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The real relation between $X$ and $Y$ can not be found out even by rigours\n",
"theory. Thus the approximation will be a good replaced solution. There are\n",
"still some problems that have to be considered:\n",
"1. Which relation do we want? The simple linear relation, i.e. $Y = m X +\n",
" b$, is a good suggestion;\n",
"- Since the relation is an approximation, the error can not be ignored.\n",
" As shown in the last picture:\n",
"$$l_1, l_2, \\cdots, l_5 \\text{ errors}$$\n",
"\n",
"The linear approximation will be a good candidate if the sum of errors\n",
"\n",
"$$ E = l_1 + \\cdots + l_5 $$\n",
" is minimum! But different signs of errors will reduce the sum of errors.\n",
" This means that the sum of errors is very small but the differences between\n",
" exact values and observed data are very large. Therefore we can consider to\n",
" minimize the sum of the square of errors:\n",
"\n",
"$$ \\color{brown}{E' = l_1^2 + \\cdots + l_5^2} $$\n",
"\n",
"The following famous theorem given by Gauss describes how to predict the\n",
"relation from the data come from the real world:\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Theorem\n",
"---\n",
" The line $l = m x + b$ that best fits the data points $(x_1, y_1), (x_2,\n",
" y_2), \\cdots, (x_n, y_n)$ is the line for which sum of the sum of square\n",
" errors\n",
" $$ E_1 + E_2 + \\cdots + E_n, \\text{ where } E_i = (y_i - m x_i - b)^2 $$\n",
" is a minimum and\n",
"\n",
"$$ m = \\frac{\\overline{x y} - \\bar{x} \\bar{y}}{\\overline{x^2} - \\bar{x}^2},\n",
" b = \\frac{\\overline{x^2} \\bar{y} - \\bar{x} \\overline{x y}}{\\overline{x^2}\n",
" - \\bar{x}^2} $$\n",
" Here, we use the notation of \"average of variable\",\n",
" $\\overline{\\color{brown}{\\cdot}}$, to represent the sum of relative\n",
" variables, for example,\n",
"\n",
"$$ \\overline{x y} = \\frac{1}{n} \\sum_{k = 1}^n x_k y_k $$\n",
"\n",
"\n",
"**Proof**\n",
"\n",
" Since the sum of total error, $E_i$, is sum of (positive)\n",
" square of errors, it must attain its minimum, i.e.\n",
"\n",
"$$ E = \\sum_{k = 1}^n E_i = \\sum_{k = 1}^n (y_i - \\color{brown}{m} x_i -\n",
" \\color{brown}{b})^2 $$\n",
" $\\min E$ exists since the last sum is sum of positive squares. Then the\n",
" critical value satisfies\n",
"\n",
"$$ 0 = \\frac{\\partial E}{\\partial m} = - 2 \\sum_{k = 1}^n x_i (y_i - m x_i -\n",
" b) $$ \n",
"\n",
"$$ 0 = \\frac{\\partial E}{\\partial b} = - 2 \\sum_{k = 1}^n (y_i - m x_i - b)$$\n",
" And these can be reduced into the following linear system equations:\n",
"\n",
"$$ m \\sum_{k = 1}^n x_k^2 + b \\sum_{k = 1}^n x_k = \\sum_{k = 1}^n y_k x_k\n",
" \\longrightarrow \\color{brown}{m} \\overline{x^2} + \\color{brown}{b}\n",
" \\bar{x} = \\overline{x y} $$\n",
"\n",
"$$m \\sum_{k = 1}^n x_k^{} + n b = \\sum_{k = 1}^n y_k \\longrightarrow\n",
" \\color{brown}{m} \\overline{x^{}} + \\color{brown}{b} = \\overline{y}$$\n",
" \n",
" And by the general procedure of solving linear system of equations:\n",
"\n",
"\\begin{eqnarray*}\n",
" m & = & \\frac{\\left|\\begin{array}{cc}\n",
" \\overline{x y} & \\bar{x}\\\\\n",
" \\overline{y} & 1\n",
" \\end{array}\\right|}{\\left|\\begin{array}{cc}\n",
" \\overline{x^2 } & \\bar{x}\\\\\n",
" \\overline{x} & 1\n",
" \\end{array}\\right|}\\\\\n",
" & = & \\frac{\\overline{x y} - \\bar{x} \\bar{y}}{\\overline{x^2} - \\bar{x}^2}\n",
" \\\\\n",
" b & = & \\frac{\\left|\\begin{array}{cc}\n",
" \\overline{x^2} & \\overline{x y}\\\\\n",
" \\overline{x} & \\bar{y}\n",
" \\end{array}\\right|}{\\left|\\begin{array}{cc}\n",
" \\overline{x^2 } & \\bar{x}\\\\\n",
" \\overline{x} & 1\n",
" \\end{array}\\right|}\\\\\n",
" & = & \\frac{\\overline{x^2} \\bar{y} - \\bar{x} \\overline{x\n",
" y}}{\\overline{x^2} - \\bar{x}^2}\n",
" \\end{eqnarray*}\n",
" \n",
" \n",
" we have the result.\n",
"\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Example\n",
"---\n",
" The discount rate (in $\\%$) for 5 months beginning in June of a given year\n",
" in U.S.A. is as following table:\n",
" \n",
"\n",
" June July August September October\n",
" 14 13 10 10 9\n",
"\n",
"1. Find the linear approximation for discount rate with respect to time\n",
" (month), i.e. the least square regression.\n",
"- Predict the discount rates in November and December.\n",
"\n",
"**Sol:** Let $x_i$ be the number of $i$-months after June with $y_i$\n",
"be the discount rate during this month, then we\n",
"have\n",
"\n",
"$$ \\sum_{i = 1}^5 x_i = 10 ; \\sum_{i = 1}^5 y_i = 56 ; \\sum_{i = 1}^5 x_i^2 =\n",
" 30 ; \\sum_{i = 1}^5 x_i y_i = 99. $$\n",
"\n",
"\n",
"**1. **\n",
"\n",
"$$y = m x + b = \\frac{5 \\cdot 99 - 10 \\cdot 56}{5 \\cdot 30^{} - 10^2} x\n",
" + \\frac{30 \\cdot 56 - 10 \\cdot 99}{5 \\cdot 30^{} - 10^2} = - \\frac{13}{10} x\n",
" + \\frac{69}{5}$$\n",
"\n",
"**2.** Since Nov. and Dec. $x'$s are 5 and 6 respectively, we have\n",
"\n",
"$$ y_{\\text{Nov}} = - \\frac{13}{10} 5 + \\frac{69}{5} = 7.3\\%;\\\\ y_{\\text{Dec}}\n",
" = - \\frac{13}{10} 6 + \\frac{69}{5} = 6\\% $$"
]
},
{
"cell_type": "code",
"execution_count": 126,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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DQztHlPRBtv3uvrdxvEpXbmY+r9IfOAmArVu3Mnjw4NzylYPZs+GMM+DP/zz7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\nDHgB+Gs//I9NcRLlHwHfPNb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