SOLUTION: 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. The first two points of the iter

Question 1199332: 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.
The first two points of the iteration are
x1=(-2.46 , -4.08 ) CORRECT
x2=(-2.13 , -4.1228 ) CORRECT
The main function is in
xopt=(952717619 , -150328908.2) WRONG
The optimal 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) = 6xy - (19x² + 3y²) - 36x - 14y + 13
* **Gradient of the Function:**
∇f(x, y) = (∂f/∂x, ∂f/∂y) = (6y - 38x - 36, 6x - 6y - 14)
**2. Implement Gradient Ascent**
* **Initialize:**
- `x0 = np.array([-3, -4])`
- `learning_rate = 0.01`
- `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 6*x[0]*x[1] - (19*x[0]**2 + 3*x[1]**2) - 36*x[0] - 14*x[1] + 13
# Define the gradient of the function
def grad_f(x):
return np.array([6*x[1] - 38*x[0] - 36, 6*x[0] - 6*x[1] - 14])
# Initial point
x0 = np.array([-3, -4])
# Learning rate
learning_rate = 0.01
# 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: [-1.56250003 -3.89583352]
Maximum value: 68.39583333333324
```
**Therefore:**
* **x_opt = (-1.5625, -3.8958)**
* **f_opt = 68.3958**
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).

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