Step-by-step explanation:
To solve the given unconstrained optimization problem using the steepest descent method, we need to find the minimum of the objective function
f(x) = (x₁ - 1)² + (x₂ - 1)²,
starting from the initial point x₀ = (0, 0).
The steepest descent method is an iterative optimization algorithm that updates the current solution by taking steps in the direction of the negative gradient. The negative gradient points in the direction of steepest descent, which is the direction of the fastest decrease of the objective function.
Step-by-step explanation:
1. Start with the initial point x₀ = (0, 0).
2. Compute the gradient ∇f(x) at the current point x.
∇f(x) = (∂f/∂x₁, ∂f/∂x₂)
= (2(x₁ - 1), 2(x₂ - 1))
= (2x₁ - 2, 2x₂ - 2)
3. Determine the step size α (also known as the learning rate). This step size determines how far we move in the direction of the negative gradient in each iteration.
4. Update the current point x using the steepest descent update rule:
x_new = x - α * ∇f(x)
= (x₁ - α(2x₁ - 2), x₂ - α(2x₂ - 2))
= (x₁(1 - 2α) + 2α, x₂(1 - 2α) + 2α)
5. Repeat steps 2-4 until convergence or a stopping criterion is met.
Note:
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