 
Summary: Preprint ANL/MCSP16410609.
ONLINE NONLINEAR PROGRAMMING AS A GENERALIZED
EQUATION
VICTOR M. ZAVALA AND MIHAI ANITESCU
Abstract. We establish results for the problem of tracking a timedependent manifold arising in on
line nonlinear programming by casting this as a generalized equation. We demonstrate that if points along
a solution manifold are consistently strongly regular, it is possible to track the manifold approximately by
solving a single linear complementarity problem (LCP) at each time step. We derive sufficient conditions
guaranteeing that the tracking error remains bounded to second order with the size of the time step, even if
the LCP is solved only to firstorder accuracy. We use these results to derive a fast, augmented Lagrangean
tracking algorithm and demonstrate the developments through a numerical case study.
Key words. generalized equations, stability, nonlinear programming, online, complementarity
AMS subject classifications. 34B15, 34H05, 49N35, 49N90, 90C06, 90C30, 90C55, 90C59
1. Introduction. Advanced online optimization, control, and estimation strategies rely
on repetitive solutions of nonlinear programming (NLP) problems. The structure of the NLP
is normally fixed, but it depends on timedependent data obtained at predefined sampling
times (e.g. sensor measurements and model states).
Traditional online NLP strategies try to extend the sampling time (time step) as much
as possible in order to accommodate the solution of the NLP to a fixed degree of accuracy.
A problem with this approach is that it neglects the fact that the NLP solver is implicitly
