To find the optimal step size $\alpha_k$ for the gradient descent scheme to minimize the function $g(z)$, we start by analyzing the function and its gradient.

Step 1: Define the function and compute the gradient

The function $g(z)$ is defined as: $g(z) = \frac{1}{2} \|Az - b\|_2^2 + \frac{1}{2} \|z\|_2^2$ Expanding the norm terms, we have: $g(z) = \frac{1}{2} (Az - b)^\top (Az - b) + \frac{1}{2} z^\top z$

Now, let’s compute the gradient of $g(z)$ with respect to $z$: $\nabla g(z) = \nabla \left( \frac{1}{2} \|Az - b\|_2^2 + \frac{1}{2} \|z\|_2^2 \right)$

For the first term: $\nabla \left( \frac{1}{2} \|Az - b\|_2^2 \right) = \nabla \left( \frac{1}{2} (Az - b)^\top (Az - b) \right)$ Using the chain rule, we have: $\nabla \left( \frac{1}{2} (Az - b)^\top (Az - b) \right) = A^\top (Az - b)$

For the second term: $\nabla \left( \frac{1}{2} \|z\|_2^2 \right) = \nabla \left( \frac{1}{2} z^\top z \right) = z$

Thus, the gradient of $g(z)$ is: $\nabla g(z) = A^\top (Az - b) + z$

Step 2: Gradient Descent Update Rule

The gradient descent update rule at iteration $k$ is: $z_k = z_{k-1} - \alpha_k \nabla g(z_{k-1})$ Substituting the gradient: $z_k = z_{k-1} - \alpha_k \left( A^\top (A z_{k-1} - b) + z_{k-1} \right)$

Step 3: Find the Optimal Step Size $\alpha_k$

To find the optimal step size $\alpha_k$, we minimize $g(z)$ along the direction of the negative gradient. Define the univariate function: $h(\alpha) = g(z_{k-1} - \alpha \nabla g(z_{k-1}))$ Differentiate $h(\alpha)$ with respect to $\alpha$ and set it to zero to find the optimal $\alpha_k$.

Substitute $z = z_{k-1} - \alpha \nabla g(z_{k-1})$ into $g(z)$: $h(\alpha) = \frac{1}{2} \|A(z_{k-1} - \alpha \nabla g(z_{k-1})) - b\|_2^2 + \frac{1}{2} \|z_{k-1} - \alpha \nabla g(z_{k-1})\|_2^2$ Expanding the norm:

+ \frac{1}{2} \left( z_{k-1} - \alpha \nabla g(z_{k-1}) \right)^\top \left( z_{k-1} - \alpha \nabla g(z_{k-1}) \right)$$ Taking the derivative with respect to $$\alpha$$ and setting it to zero: $$\frac{d h(\alpha)}{d \alpha} = -\nabla g(z_{k-1})^\top \left( A^\top (Az_{k-1} - b) + z_{k-1} \right) + \alpha \nabla g(z_{k-1})^\top \left( A^\top A + I \right) \nabla g(z_{k-1}) = 0$$ Thus, the optimal step size $$\alpha_k$$ is: $$\alpha_k = \frac{\nabla g(z_{k-1})^\top \nabla g(z_{k-1})}{\nabla g(z_{k-1})^\top (A^\top A + I) \nabla g(z_{k-1})}$$ This expression provides the optimal step size $$\alpha_k$$ for each iteration in the gradient descent method. Would you like more details on any part of the derivation, or do you have any questions? Here are 5 related questions you might find interesting: 1. How does the convergence rate of gradient descent change with different step sizes? 2. Can the optimal step size $$\alpha_k$$ be constant for all iterations, and under what conditions? 3. How does the matrix $$A$$ affect the optimization problem and the gradient descent convergence? 4. What are the implications of choosing a step size $$\alpha_k$$ that is too large or too small? 5. How would you apply a line search method to determine the step size in a more general setting? **Tip:** When implementing gradient descent, it’s important to monitor the objective function's value at each step to ensure the algorithm is converging as expected.