Target-Following for Nonlinear Programming
For convex nonlinear programming (NLP) Nesterov and Nemirovskii  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  and Jarre , 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  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  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.
KeywordsVariational Inequality Barrier Function Nonlinear Programming Interior Point Method Central Path
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