Gradient and steepest descent methods - quadratic problem

Question

Gradient and steepest descent methods

1. Quadratic problem in R². Verify the expressions for the iterates x^(k).

2. Let Δx_sd and Δx_sd be the normalized and unnormalized steepest descent directions at x, for the norm ||.||. Prove the following identities.

(a) ∇ f(x)^TΔx_nsd = -||∇f(x)||_*.

(b) ∇ f(x)^TΔx_sd = -||∇f(x)||²_*.

(c) Δx_sd = argmint,(∇f (x)^Tv + (1/2)||v||²).

3. Steepest descent method in l_∞-norm. Explain how to find a steepest descent direction in the l_∞-norm, and give a simple interpretation.