Asymptotic KKT Conditions for Continuous-Time Nonlinear Programming

TL;DR

This paper introduces asymptotic KKT conditions for continuous-time nonlinear programming, providing necessary and sufficient optimality conditions and proposing an augmented Lagrangian method with convergence analysis.

Contribution

It establishes asymptotic KKT conditions for continuous-time problems and develops a new augmented Lagrangian method with proven convergence.

Findings

Sequential KKT conditions are necessary for optimality.

Under convexity, these conditions are also sufficient.

The proposed method effectively solves continuous-time problems in literature.

Abstract

This paper addresses the class of continuous-time nonlinear programming problems with equality and inequality constraints. The paper presents necessary optimality conditions of the sequential form. To be more precise, a sequence of solutions converging to the optimal solution is demonstrated to exist, and such that Karush-Kuhn-Tucker-type conditions are satisfied asymptotically. It is shown that these sequential Karush-Kuhn-Tucker-type conditions also become sufficient for optimality under convexity assumptions. Sequential optimality conditions are a valuable tool for determining when to terminate a numerical method of solution. In this regard, an augmented Lagrangian-type method is proposed for numerically solving continuous-time programming problems. A convergence analysis concerning viability and optimality is presented. The performance of the method is evaluated by applying it to solve instances of continuous-time problems found in the literature.