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).
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