document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199331>Question 1199331</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the gradient method to find the maximum of the function f(x,y)=54y−(5x^2+9y^2)+70x−319 with initial point x0=(8,5) and λ=0.08. (The number λ is also known as step size or learning rate.)\r<BR>\n" );
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document.write( "The first two points of the iteration are<BR>\n" );
document.write( "x1=(7.2,2.12) CORRECT<BR>\n" );
document.write( "x2=(-2.40032,37.2096) WRONG\r<BR>\n" );
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document.write( "The maximum of the function is in (you may need to change the value of λ to achieve convergence):\r<BR>\n" );
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document.write( "xopt=(___,___)\r<BR>\n" );
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document.write( "The maximum value of the function is\r<BR>\n" );
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document.write( "fopt=_____<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848223 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848223\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Define the Function and its Gradient**\r<BR>\n" );
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document.write( "* **Function:** <BR>\n" );
document.write( "   f(x, y) = 54y - (5x² + 9y²) + 70x - 319\r<BR>\n" );
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document.write( "* **Gradient of the Function:** <BR>\n" );
document.write( "   ∇f(x, y) = (∂f/∂x, ∂f/∂y) = (70 - 10x, 54 - 18y)\r<BR>\n" );
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document.write( "**2. Implement Gradient Ascent**\r<BR>\n" );
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document.write( "* **Initialize:**<BR>\n" );
document.write( "   - `x0 = np.array([8, 5])` <BR>\n" );
document.write( "   - `learning_rate = 0.08` <BR>\n" );
document.write( "   - `max_iter = 1000` <BR>\n" );
document.write( "   - `tol = 1e-6` \r<BR>\n" );
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document.write( "* **Iterate:**<BR>\n" );
document.write( "   1. Calculate the gradient at the current point `x`.<BR>\n" );
document.write( "   2. Update `x` using the gradient ascent update rule: <BR>\n" );
document.write( "      `x = x + learning_rate * gradient`<BR>\n" );
document.write( "   3. Check for convergence (e.g., if the magnitude of the gradient is below the tolerance).\r<BR>\n" );
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document.write( "**3. Find the Maximum**\r<BR>\n" );
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document.write( "* Run the gradient ascent algorithm.<BR>\n" );
document.write( "* The final value of `x` after convergence will be the approximate location of the maximum.<BR>\n" );
document.write( "* Evaluate the function `f(x)` at this point to find the maximum value.\r<BR>\n" );
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document.write( "**4. Adjust Learning Rate (if necessary)**\r<BR>\n" );
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document.write( "* If the algorithm doesn't converge or oscillates, try adjusting the `learning_rate`. <BR>\n" );
document.write( "    * A smaller learning rate can help with convergence but may slow down the process.<BR>\n" );
document.write( "    * A larger learning rate can speed up convergence but may cause the algorithm to overshoot the maximum.\r<BR>\n" );
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document.write( "**Python Implementation**\r<BR>\n" );
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document.write( "```python<BR>\n" );
document.write( "import numpy as np\r<BR>\n" );
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document.write( "def gradient_ascent(f, grad_f, x0, learning_rate, max_iter=1000, tol=1e-6):<BR>\n" );
document.write( "  \"\"\"<BR>\n" );
document.write( "  Performs gradient ascent to find the maximum of a function.\r<BR>\n" );
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document.write( "  Args:<BR>\n" );
document.write( "    f: The function to optimize.<BR>\n" );
document.write( "    grad_f: The gradient of the function.<BR>\n" );
document.write( "    x0: The initial point.<BR>\n" );
document.write( "    learning_rate: The step size for the gradient ascent.<BR>\n" );
document.write( "    max_iter: The maximum number of iterations.<BR>\n" );
document.write( "    tol: The tolerance for convergence.\r<BR>\n" );
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document.write( "  Returns:<BR>\n" );
document.write( "    x_opt: The optimal point found by the algorithm.<BR>\n" );
document.write( "    f_opt: The maximum value of the function at x_opt.<BR>\n" );
document.write( "  \"\"\"\r<BR>\n" );
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document.write( "  x = np.array(x0)<BR>\n" );
document.write( "  for _ in range(max_iter):<BR>\n" );
document.write( "    gradient = grad_f(x)<BR>\n" );
document.write( "    x = x + learning_rate * gradient<BR>\n" );
document.write( "    if np.linalg.norm(gradient) < tol:<BR>\n" );
document.write( "      break\r<BR>\n" );
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document.write( "  return x, f(x)\r<BR>\n" );
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document.write( "# Define the function<BR>\n" );
document.write( "def f(x):<BR>\n" );
document.write( "  return 54*x[1] - (5*x[0]**2 + 9*x[1]**2) + 70*x[0] - 319\r<BR>\n" );
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document.write( "# Define the gradient of the function<BR>\n" );
document.write( "def grad_f(x):<BR>\n" );
document.write( "  return np.array([70 - 10*x[0], 54 - 18*x[1]])\r<BR>\n" );
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document.write( "# Initial point<BR>\n" );
document.write( "x0 = np.array([8, 5])\r<BR>\n" );
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document.write( "# Learning rate<BR>\n" );
document.write( "learning_rate = 0.08\r<BR>\n" );
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document.write( "# Perform gradient ascent<BR>\n" );
document.write( "x_opt, f_opt = gradient_ascent(f, grad_f, x0, learning_rate)\r<BR>\n" );
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document.write( "# Print the results<BR>\n" );
document.write( "print(f\"Maximum point: {x_opt}\")<BR>\n" );
document.write( "print(f\"Maximum value: {f_opt}\") <BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**Output:**\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "Maximum point: [7.         2.99999999]<BR>\n" );
document.write( "Maximum value: 7.0<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**Therefore:**\r<BR>\n" );
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document.write( "* **x_opt = (7.0, 3.0)** <BR>\n" );
document.write( "* **f_opt = 7.0**\r<BR>\n" );
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document.write( "This result indicates that the maximum of the function f(x, y) is approximately 7.0, which occurs at the point (7.0, 3.0).<BR>\n" );
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