SOLUTION: 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

Question 1199331: 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.)
The first two points of the iteration are
x1=(7.2,2.12) CORRECT
x2=(-2.40032,37.2096) WRONG
The maximum of the function is in (you may need to change the value of λ to achieve convergence):
xopt=(___,___)
The maximum value of the function is
fopt=_____

Answer by textot(100) (Show Source): You can put this solution on YOUR website!
**1. Define the Function and its Gradient**
* **Function:**
f(x, y) = 54y - (5x² + 9y²) + 70x - 319
* **Gradient of the Function:**
∇f(x, y) = (∂f/∂x, ∂f/∂y) = (70 - 10x, 54 - 18y)
**2. Implement Gradient Ascent**
* **Initialize:**
- `x0 = np.array([8, 5])`
- `learning_rate = 0.08`
- `max_iter = 1000`
- `tol = 1e-6`
* **Iterate:**
1. Calculate the gradient at the current point `x`.
2. Update `x` using the gradient ascent update rule:
`x = x + learning_rate * gradient`
3. Check for convergence (e.g., if the magnitude of the gradient is below the tolerance).
**3. Find the Maximum**
* Run the gradient ascent algorithm.
* The final value of `x` after convergence will be the approximate location of the maximum.
* Evaluate the function `f(x)` at this point to find the maximum value.
**4. Adjust Learning Rate (if necessary)**
* If the algorithm doesn't converge or oscillates, try adjusting the `learning_rate`.
* A smaller learning rate can help with convergence but may slow down the process.
* A larger learning rate can speed up convergence but may cause the algorithm to overshoot the maximum.
**Python Implementation**
```python
import numpy as np
def gradient_ascent(f, grad_f, x0, learning_rate, max_iter=1000, tol=1e-6):
"""
Performs gradient ascent to find the maximum of a function.
Args:
f: The function to optimize.
grad_f: The gradient of the function.
x0: The initial point.
learning_rate: The step size for the gradient ascent.
max_iter: The maximum number of iterations.
tol: The tolerance for convergence.
Returns:
x_opt: The optimal point found by the algorithm.
f_opt: The maximum value of the function at x_opt.
"""
x = np.array(x0)
for _ in range(max_iter):
gradient = grad_f(x)
x = x + learning_rate * gradient
if np.linalg.norm(gradient) < tol:
break
return x, f(x)
# Define the function
def f(x):
return 54*x[1] - (5*x[0]**2 + 9*x[1]**2) + 70*x[0] - 319
# Define the gradient of the function
def grad_f(x):
return np.array([70 - 10*x[0], 54 - 18*x[1]])
# Initial point
x0 = np.array([8, 5])
# Learning rate
learning_rate = 0.08
# Perform gradient ascent
x_opt, f_opt = gradient_ascent(f, grad_f, x0, learning_rate)
# Print the results
print(f"Maximum point: {x_opt}")
print(f"Maximum value: {f_opt}")
```
**Output:**
```
Maximum point: [7. 2.99999999]
Maximum value: 7.0
```
**Therefore:**
* **x_opt = (7.0, 3.0)**
* **f_opt = 7.0**
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).

RELATED QUESTIONS
Use the gradient rise method©to optimize the function... (answered by textot)

Derivative of Function using the given formula: F^1(X0)= Lim x-XO (fx) - f(XO)/x-X0 (answered by Alan3354)

Hi Can Someone please help Use the Newton-Raphson method to find a solution to four... (answered by stanbon)

write an equation of the conic section parabola with vertex at (2,6) and directrix at... (answered by lwsshak3)

Please Help Me With This Question find the gradient of the curve y=x^2+5x-3 at the point... (answered by Boreal)

Use the following function to find f(8) and f(0)... (answered by macston)

having problems/please help 1.find the x-intercept and y-intercept of the equation... (answered by checkley75)

Just looking for a check to make sure I'm on the right track. Thanks 35. Given a line... (answered by josgarithmetic,MathTherapy)

The function f(x0 = -x^2+46x-360 models the daily profit in hundreds of dollars for a... (answered by ewatrrr)