Dual Newton method. In this problem we explore Newton's method for solving the dual of the equality constrained minimization problem (10.1). We assume that f is twice differentiable, ∇2f(x)≻0 for all x∈domf, and that for each ν∈Rp, the Lagrangian L(x,ν)=f(x)+νT(Ax−b) has a unique minimizer, which we denote x(ν).
((a)) Show that the dual function g is twice differentiable. Find an expression for the Newton step for the dual function g, evaluated at ν, in terms of f, ∇f, and ∇2f, evaluated at x=x(ν). You can use the results of exercise 3.40.
((b)) Suppose there exists a K such that
[∇2f(x)AAT0]−12≤K
for all x∈domf. Show that g is strongly concave, with ∇2g(ν)⪯−(1/K)I.