Target-Following for Nonlinear Programming

Abstract

For convex nonlinear programming (NLP) Nesterov and Nemirovskii [193] gave an analysis of the standard (primal) short-step logarithmic barrier method. They introduced the self-concordance condition (Definition 6.5.3), which generalizes earlier smoothness conditions by Zhu [258] and Jarre [118], and is generally accepted as the one suitable for the analysis of short-step methods. In this chapter we show that the target-following framework can be applied to convex programming problems as well, thereby generalizing the analysis of Nesterov and Nemirovskii [193] to interior point methods not necessarily following the central path, see Section 9.2. Here, we use the fact that the primal-dual system (8.1) is also the KKT-system for minimizing the primal (or dual) barrier function over the primal (dual) feasible region. The main difference of our analysis compared to the one in [193] is that in our case the self-concordance parameters are not constant but change from one iteration to another, depending on the change in the targets. As far as we know, many of the methods we analyze in this section have been analyzed and applied to LP, but not analyzed for NLP problems.