Convex Optimization · Problem 10.7

Interior-point methods — Boyd & Vandenberghe

Mathematics — Optimization · assumptions-infeasible-start-newton

Problem

Assumptions for infeasible start Newton method. Consider the set of assumptions given on page 536. ((a)) Suppose that the function ff is closed. Show that this implies that the norm of the residual, ∥r(x,ν)∥2\|r(x, \nu)\|_2, is closed. ((b)) Show that DrDr satisfies a Lipschitz condition if and only if ∇2f\nabla^2 f does.
Intuitive explanation

The infeasible start Newton method solves constrained optimization by tracking both primal and dual variables through a residual vector r(x,ν)r(x, \nu). Part (a) asks you to translate the closedness property of the objective function ff into a property of the residual norm—think about how closedness (level sets are closed) propagates through composition with the norm. Part (b) connects smoothness of the Hessian to smoothness of the Jacobian of the residual: since rr is built from ff's derivatives, the Lipschitz constant of DrDr should directly reflect the Lipschitz constant of ∇2f\nabla^2 f.

Intuitive explanation

The infeasible start Newton method solves constrained optimization by tracking both primal and dual variables through a residual vector r(x,ν)r(x, \nu). Part (a) asks you to translate the closedness property of the objective function ff into a property of the residual norm—think about how closedness (level sets are closed) propagates through composition with the norm. Part (b) connects smoothness of the Hessian to smoothness of the Jacobian of the residual: since rr is built from ff's derivatives, the Lipschitz constant of DrDr should directly reflect the Lipschitz constant of ∇2f\nabla^2 f.

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