Primal-Dual Interior Methods for Nonlinear Optimization

Regularization and stabilization are vital tools for resolving the
numerical and theoretical difficulties associated with ill-posed or
degenerate optimization problems. Broadly speaking, regularization
involves perturbing the underlying linear equations so that they are always
nonsingular. Stabilized methods are designed to provide a sequence of
iterates with fast local convergence even when the gradients of the
constraints satisfied at a solution are linearly dependent.

We discuss the crucial role of regularization and stabilization in the
formulation and analysis of modern interior methods for nonlinear
optimization. In particular, we establish the close relationship between
regularization and stabilization, and propose a new interior method based
on formulating an associated ``simpler" optimization subproblem defined in
terms of both primal and dual variables.