document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199332>Question 1199332</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Use the gradient rise method©to optimize the function f(x,y)=6xy−(19x^2+3y^2)−36x−14y+13 with starting point x0=(−3,−4) and λ=0.01.\r<BR>\n" );
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document.write( "The first two points of the iteration are\r<BR>\n" );
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document.write( "x1=(-2.46 , -4.08 ) CORRECT<BR>\n" );
document.write( "x2=(-2.13 , -4.1228 ) CORRECT\r<BR>\n" );
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document.write( "The main function is in\r<BR>\n" );
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document.write( "xopt=(952717619 , -150328908.2) WRONG\r<BR>\n" );
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document.write( "The optimal 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 #848217 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=848217\">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) = 6xy - (19x² + 3y²) - 36x - 14y + 13\r<BR>\n" );
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document.write( "* **Gradient of the Function:** <BR>\n" );
document.write( "   ∇f(x, y) = (∂f/∂x, ∂f/∂y) = (6y - 38x - 36, 6x - 6y - 14)\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([-3, -4])` <BR>\n" );
document.write( "   - `learning_rate = 0.01` <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 6*x[0]*x[1] - (19*x[0]**2 + 3*x[1]**2) - 36*x[0] - 14*x[1] + 13\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([6*x[1] - 38*x[0] - 36, 6*x[0] - 6*x[1] - 14])\r<BR>\n" );
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document.write( "# Initial point<BR>\n" );
document.write( "x0 = np.array([-3, -4])\r<BR>\n" );
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document.write( "# Learning rate<BR>\n" );
document.write( "learning_rate = 0.01\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: [-1.56250003 -3.89583352]<BR>\n" );
document.write( "Maximum value: 68.39583333333324<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 = (-1.5625, -3.8958)** <BR>\n" );
document.write( "* **f_opt = 68.3958**\r<BR>\n" );
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document.write( "This result indicates that the maximum of the function f(x, y) is approximately 68.3958, which occurs at the point (-1.5625, -3.8958).<BR>\n" );
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