Differential-Algebraic Equations and Their Connections to Optimization

Victoria Conference Centre,

Victoria, British Columbia, Canada

**Organizers: H. Georg Bock, Johannes P. Schloder and
Volker H. Schulz, Interdisciplinary Center for Scientific Computing (IWR), University of Heidelberg, Germany**

DAE are connected to optimization in at least two ways. On the one hand variational principles are used to formulate DAE, e.g. in multibody dynamics or in boundary value problems associated with optimal control problems. This results in special structures of the differential algebraic equations. On the other hand many relevant practical problems, which are modelled by DAE, call for optimization rather than forward simulation alone. The special structures of the optimization problems arising from discretization of the DAE are investigated. It is shown how these structures can - and must - be exploited for the design of efficient optimization methods.

**Session 1** recalls why many relevant dynamical processes in
science and industry are favourably modelled by DAE and explains
the typical properties of such models (index, invariants,
implicitness, discontinuities). Challenging classes of
optimization problems parameter estimation, optimum experiment
design and path constrained optimal control problems
are formulated. Their demands on methods for discretization and
optimization are described.

**Session 2** reviews onestep and multistep discretization methods
for DAE. Emphasis is laid on advanced BDFmethods. It is shown
how such methods can be designed to optimally cooperate with
optimization algorithms.This includes solution of systems with
relaxed algebraic constraints and efficient generation of
derivatives for the optimization procedure even in the case
of DAE with implicitly characterized discontinuities.

**Session 3** is devoted to the Optimization Boundary Value Problem
Approach. A class of efficient numerical methods is presented,
which solve problems with DAE boundary value problems as
nonlinear constraint. Adequate numerical formulations of these problems are given, that help to transfer the inherent structures of DAE optimization problems so that they can be exploited in the largescale optimization methods. For the discretization of DAE boundary value problems multiple shooting and collocation methods are described and compared. These methods take advantage of invariants to improve the conditioning. For the treatment of the resulting large scale constrained nonlinear optimization problems structure exploiting SQPtype methods, that have proven to be very effective in practical applications, are discussed.
These are GaussNewton Methods for discretized parameter estimation problems and structured SQPmethods using high rank updates or partially reduced SQPmethods for constrained optimal control problems and optimal design problems.

The first part of** session 4** concentrates on techniques for the
evaluation and solution of quadratic subproblems, which are
computationally most expensive. Techniques are described
that allow to generate the functions and derivatives with an
accuracy that just meets the demands of the optimization procedure in order to reduce the computational load. Structure exploiting recursive techniques for the solution of the quadratic subproblems are given. The methods offer a high potential of parallelism on several levels, which can be exploited for further acceleration. New parallel algorithms are described and performance results are reported. Finally, a thorough discussion of several practical applications with typical numerical challenges is given. These problems include identification of mechanical systems, optimum experiment design for estimation of dynamical parameters in models for industrial robots, optimal trajectories for satellite mounted robots and chemical engineering applications in combustion.

**H. Georg Bock** holds a chair for Scientific Computing and Optimization. University of Heidelberg. In 1986 he received his Ph.D. in Mathematics from the University of Bonn. In 1988 he got a professorship at the University of Augsburg and moved to Heidelberg 1991. His main interests include the optimal combination of discretization and optimization methods and the solution of nonlinear optimal control and feedback problems.

**Johannes P. Schloder** is a senior scientist at IWR. Since 1987 he holds a Ph.D. in Mathematics from the University of Bonn and had teaching and research positions at the Universities of Bonn and Augsburg. Dr. Schloder's current research aereas are parameter estimation and optimum experiment design.

**Volker H. Schulz** completed his Ph.D. in Mathematics this year.His main interest is the development of partially reduced SQP-methods for the treatment of constrained optimal control problems in DAE and the combination of SQP-methods with multigrid procedures for the optimization of PDE.

8:00 | Registration |

9:00-10:00 | Session 1: Introduction to Optimization Connections to DAE. |

- Features of DAE Models in Applications | |

- Parameter Estimation, Optimum Experiment Design, Optimal Control Problems in DAE | |

10:00-10:30 | Coffee |

10:30-12:00 | Session 2: Discretization Methods for DAE. |

- BDF Methods and Alternative Discretizations | |

- Higher Index Problems and Invariants | |

- Treatment of Inconsistent Initial Values and Discontinuities | |

- Linear Algebra Techniques for the Reduction of the Overall Effort |

12:00-1:30 | Lunch (attendees are on their own for lunch) |

1:30-3:00 | Session 3: Efficient Treatment of Optimization Problems in DAE. |

- The Boundary Value Problem Approach | |

- Multiple Shooting and Collocation Discretization | |

- Generalized Gauss Newton Methods | |

- Structured and Partially Reduced SQPMethods | |

3:00-3:30 | Coffee |

3:30-5:30 | Session 4: Further Algorithmic Features and Practical Applications. |

- Parallel Evaluation and Solution of Quadratic Subproblems | |

- Internal Differentiation for Efficient Derivative Evaluation in Adaptive Discretization Applications from, e.g., | |

- Mechanical Engineering | |

- Robotics | |

- Chemical Engineering | |

- Enviromental Physics | |

5:30 | Short Course adjourns |

SIAG/Opt Member* | SIAM Member | Non-Member | Student | |
---|---|---|---|---|

Preregistration (before 5/6/96) | $110 | $110 | $125 | $40 |

Registration (after 5/6/96) | $125 | $125 | $140 | $55 |

**Member of SIAM Activity Group on Optimization.*

Short Course fees include course notes and refreshment breaks. To register for either short course, the conference, or both, please fill-in and submit the preregistration form.

On-site registration will start on Saturday, May 18 at 6:00 PM at the entrance, Lobby Level of the Conference Centre.