Regularizing Bilevel Nonlinear Programs by Lifting

This paper considers a bilevel nonlinear program (NLP) whose lower-level problem satisfies a linear independence constraint qualification (LICQ) and a strong second-order condition (SSOC). One would expect the resulting mathematical program with complementarity constraints (MPCC), whose constraints are the first-order optimality conditions of the lower-level NLP, to satisfy an MPEC-LICQ. We provide an example which … Read more

Efficient Direct Multiple Shooting for Nonlinear Model Predictive Control on Long Horizons

We address direct multiple shooting based algorithms for nonlinear model predictive control, with a focus on problems with long prediction horizons. We describe different efficient multiple shooting variants with a computational effort that is only linear in the horizon length. Proposed techniques comprise structure exploiting linear algebra on the one hand, and approximation of derivative … Read more

Reliable solution of convex quadratic programs with parametric active set methods

Parametric Active Set Methods (PASM) are a relatively new class of methods to solve convex Quadratic Programming (QP) problems. They are based on tracing the solution along a linear homotopy between a QP with known solution and the QP to be solved. We explicitly identify numerical challenges in PASM and develop strategies to meet these … Read more

Newton–Picard-Based Preconditioning for Linear-Quadratic Optimization Problems with Time-Periodic Parabolic PDE Constraints

We develop and investigate two preconditioners for a basic linear iterative splitting method for the numerical solution of linear-quadratic optimization problems with time-periodic parabolic PDE constraints. The resulting real-valued linear system to be solved is symmetric indefinite. We propose all-at-once symmetric indefinite preconditioners based on a Newton–Picard approach which divides the variable space into slow … Read more

A Factorization with Update Procedures for a KKT Matrix Arising in Direct Optimal Control

Quadratic programs obtained for optimal control problems of dynamic or discrete–time processes usually involve highly block structured Hessian and constraints matrices. Efficient numerical methods for the solution of such QPs have to respect and exploit this block structure. In interior point methods, this is elegantly achieved by the widespread availability of advanced sparse symmetric indefinite … Read more

Block Structured Quadratic Programming for the Direct Multiple Shooting Method for Optimal Control

In this contribution we address the efficient solution of optimal control problems of dynamic processes with many controls. Such problems arise, e.g., from the outer convexification of integer control decisions. We treat this optimal control problem class using the direct multiple shooting method to discretize the optimal control problem. The resulting nonlinear problems are solved … Read more