Conjugate of a self-concordant function. Suppose f:Rn→R is closed, strictly convex, and self-concordant. We show that its conjugate (or Legendre transform) f∗ is self-concordant.
((a)) Show that for each y∈domf∗, there is a unique x∈domf that satisfies y=∇f(x). Hint. Refer to the result of exercise 9.19.
((b)) Suppose yˉ=∇f(xˉ). Define
g(t)=f(xˉ+tv),h(t)=f∗(yˉ+tw)
where v∈Rn and w=∇2f(xˉ)v. Show that
g′′(0)=h′′(0),g′′′(0)=−h′′′(0).
Use these identities to show that f∗ is self-concordant.