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To solve this optimization problem using the **steepest descent method**, we'll follow a systematic approach. Here's the given function and the starting point:

### Given:
\[ f(x_1, x_2) = x_1 - x_2 + 2x_1^2 + 2x_1x_2 + x_2^2 \]
- Starting point: \( (x_1, x_2) = (0, 0) \)

### Steps for the Steepest Descent Method:

1. **Compute the gradient** of the function \( f(x_1, x_2) \).
2. **Evaluate the gradient at the starting point**.
3. **Determine the search direction** as the negative gradient.
4. **Perform a line search** to find the optimal step size along the search direction.
5. **Update the point** and repeat the process until convergence.

Let's go through these steps.

### Step 1: Compute the Gradient
The gradient \( \nabla f(x_1, x_2) \) consists of the partial derivatives with respect to \( x_1 \) and \( x_2 \):

\[
\frac{\partial f}{\partial x_1} = 1 + 4x_1 + 2x_2
\]
\[
\frac{\partial f}{\partial x_2} = -1 + 2x_1 + 2x_2
\]

Thus, the gradient is:
\[
\nabla f(x_1, x_2) = \left( 1 + 4x_1 + 2x_2, -1 + 2x_1 + 2x_2 \right)
\]

### Step 2: Evaluate the Gradient at the Starting Point \( (0, 0) \)
At the point \( (x_1, x_2) = (0, 0) \):
\[
\nabla f(0, 0) = \left( 1 + 4(0) + 2(0), -1 + 2(0) + 2(0) \right) = (1, -1)
\]

### Step 3: Determine the Search Direction
The search direction \( p \) is the negative of the gradient:
\[
p = -\nabla f(0, 0) = (-1, 1)
\]

### Step 4: Perform a Line Search
We need to find the optimal step size \( \alpha \) that minimizes \( f(x_1, x_2) \) along the direction \( p = (-1, 1) \). 

Let the new point be:
\[
(x_1, x_2) = (0, 0) + \alpha(-1, 1) = (-\alpha, \alpha)
\]
Substituting into the function \( f(x_1, x_2) \):
\[
f(-\alpha, \alpha) = -\alpha - \alpha + 2(-\alpha)^2 + 2(-\alpha)(\alpha) + \alpha^2
\]
Simplifying:
\[
f(-\alpha, \alpha) = -2\alpha + 2\alpha^2 - 2\alpha^2 + \alpha^2 = -2\alpha + \alpha^2
\]
To minimize this with respect to \( \alpha \), we take the derivative:
\[
\frac{d}{d\alpha}(-2\alpha + \alpha^2) = -2 + 2\alpha
\]
Setting this equal to 0:
\[
-2 + 2\alpha = 0 \quad \Rightarrow \quad \alpha = 1
\]

### Step 5: Update the Point
Using \( \alpha = 1 \), the new point is:
\[
(x_1, x_2) = (0, 0) + 1(-1, 1) = (-1, 1)
\]

Now, we can repeat the process starting from the new point \( (-1, 1) \). The iterations continue until the gradient is close to zero or the change in the function value is sufficiently small.

Do you want to continue with additional iterations, or do you have any questions?

Here are 5 relative questions:
1. How does the steepest descent method compare with Newton's method in terms of convergence speed?
2. What are the potential issues with using a fixed step size in the steepest descent method?
3. How can we modify the steepest descent method to improve convergence?
4. What role does the gradient play in optimization problems?
5. How can you determine when to stop the iteration process in steepest descent?

**Tip**: In optimization, choosing the correct step size is crucial; adaptive methods like line search can improve convergence.

Solve the function f(x1, x2) = x1 - x2 + 2x1^2 + 2x1x2 + x2^2 using the steepest descent method. This problem involves key optimization concepts such as gradient descent, line search, and convergence criteria. Ideal for college-level students learning calculus-based optimization methods.

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