Relations Between Abs-Normal NLPs and MPCCs Part 2: Weak Constraint Qualifications

This work continues an ongoing effort to compare non-smooth optimization problems in abs-normal form to Mathematical Programs with Complementarity Constraints (MPCCs). We study general Nonlinear Programs with equality and inequality constraints in abs-normal form, so-called Abs-Normal NLPs, and their relation to equivalent MPCC reformulations. We introduce the concepts of Abadie’s and Guignard’s kink qualification and … Read more

Relations Between Abs-Normal NLPs and MPCCs Part 1: Strong Constraint Qualifications

This work is part of an ongoing effort of comparing non-smooth optimization problems in abs-normal form to MPCCs. We study the general abs-normal NLP with equality and inequality constraints in relation to an equivalent MPCC reformulation. We show that kink qualifications and MPCC constraint qualifications of linear independence type and Mangasarian-Fromovitz type are equivalent. Then … Read more

Efficient Derivative Evaluation for Rigid-body Dynamics based on Recursive Algorithms subject to Kinematic and Loop Constraints

Simulation, optimization and control of robotic and bio-mechanical systems depend on a mathematical model description, typically a rigid-body system connected by joints, for which efficient algorithms to compute the forward or inverse dynamics exist. Models that e.g.\ include spring-damper systems are subject to both kinematic and loop constraints. Gradient-based optimization and control methods require derivatives … Read more

A switching cost aware rounding method for relaxations of mixed-integer optimal control problems

This article investigates a class of Mixed-Integer Optimal Control Problems (MIOCPs) with switching costs. We introduce the problem class of Minimal-Switching-Cost Optimal Control Problems (MSCP) with an objective function that consists of two summands, a continuous term depending on the state vector and an encoding of the discrete switching costs. State vectors of Mixed-Integer Optimal … Read more

Numerical Solution of Optimal Control Problems with Switches, Switching Costs and Jumps

In this article, we present a framework for the numerical solution of optimal control problems, constrained by ordinary differential equations which can run in (finitely many) different modes, where a change of modes leads to additional switching cost in the cost function, and whenever the system changes its mode, jumps in the differential states are … Read more

On the Relation between MPECs and Optimization Problems in Abs-Normal Form

We show that the problem of unconstrained minimization of a function in abs-normal form is equivalent to identifying a certain stationary point of a counterpart Mathematical Program with Equilibrium Constraints (MPEC). Hence, concepts introduced for the abs-normal forms turn out to be closely related to established concepts in the theory of MPECs. We give a … Read more

Improved Regularity Assumptions for Partial Outer Convexification of Mixed-Integer PDE-Constrained Optimization problems

Partial outer convexification is a relaxation technique for MIOCPs being constrained by time-dependent differential equations. Sum-Up-Rounding algorithms allow to approximate feasible points of the relaxed, convexified continuous problem with binary ones that are feasible up to an arbitrarily small $\delta > 0$. We show that this approximation property holds for ODEs and semilinear PDEs under … Read more

Approximation Properties of Sum-Up Rounding in the Presence of Vanishing Constraints

Approximation algorithms like sum-up rounding that allow to compute integer-valued approximations of the continuous controls in a weak$^*$ sense have attracted interest recently. They allow to approximate (optimal) feasible solutions of continuous relaxations of mixed-integer control problems (MIOCPs) with integer controls arbitrarily close. To this end, they use compactness properties of the underlying state equation, … Read more

Combining Multi-Level Real-time Iterations of Nonlinear Model Predictive Control to Realize Squatting Motions on Leo

Today’s humanoid robots are complex mechanical systems with many degrees of freedom that are built to achieve locomotion skills comparable to humans. In order to synthesize whole-body motions, real-tme capable direct methods of optimal control are a subject of contemporary research. To this end, Nonlinear Model Predictive Control is the method of choice to realize … Read more

High-Level Interfaces for the Multiple Shooting Code for Optimal Control MUSCOD

The demand for model-based simulation and optimization solutions requires the availability of software frameworks that not only provide computational capabilities, but also help to ease the formulation and implementation of the respective optimal control problems. In this article, we present and discuss recent development efforts and applicable work flows using the example of MUSCOD, the … Read more