Constrained Nonlinear Programming

Abstract

We have already briefly touched nonlinear programming in the last two chapters, in which we introduced the multiple shooting discretized optimal control problem, a Nonlinear Program (NLP), and presented the outer convexification and relaxation approach that allows to compute approximations to local Mixed–Integer Optimal Control Problem (MIOCP) solutions by solving a reformulated and discretized but possibly much larger Optimal Control Problem (OCP). This chapter is concerned with theory and numerical methods for the solution of NLPs and equips the reader with definitions and algorithms required for the following chapters of this thesis. We present optimality conditions characterizing locally optimal solutions and introduce Sequential Quadratic Programming (SQP) methods for the iterative solution of NLPs. The evaluation of the matching condition constraints of the discretized optimal control problem requires the solution of Initial Value Problems (IVPs). To this end, we present one step methods for non–stiff Ordinary Differential Equations (ODEs) and discuss the efficient and consistent computation of sensitivities of IVPs solutions. The familiar reader may wish to continue with chapter 4 rightaway, and refer back to this chapter should the need arise.