### Optimization and Systems Theory Seminar

Tuesday, September 25, 2001, 11.00-12.00, Room 3721, Lindstedtsvägen 25

**Sven Leyffer**

Department of Mathematics

University of Dundee

Dundee, Scotland, UK

E-mail:
sleyffer@maths.dundee.ac.uk

####
How the Grinch solved MPECs, mathematical programs with
equilibrium constraints

Equilibrium constraints in the form of complementarity conditions, and
more generally variational inequalities, often appear as constraints
in optimization problems. Applications of equilibrium constraints are
widespread and fast growing. They cover very diverse areas such as the
design of structures involving friction, elasto-hydrodynamic
lubrication, taxation models, the modeling of competition in
deregulated electricity markets and transportation network design.
Over recent years, it has become evident that equilibrium constraints
cannot be solved satisfactorily with standard techniques for Nonlinear
Programming (NLP). Both numerical and theoretical evidence has been
advanced which support this view.

This talk is aimed at a general mathematical audience and starts by
introducing and reviewing equilibrium constraints. We then give some
applications which emphasize the usefulness and elegance of
equilibrium constraints as a modeling tool.

Next, the assertion that standard techniques for NLP cannot be applied
to equilibrium constraints is re-examined and some startling numerical
evidence is presented using our own NLP solver.

The talk concludes by examining the local convergence properties of
certain NLP methods applied to MPECs. It is shown that a simple
constraint relaxation strategy allows a proof of second order
convergence to be given under reasonable assumptions. A number of
illustrative examples are presented which show that some of the
assumptions are difficult to relax.

Calendar of seminars

*Last update: September 14, 2001 by
Anders Forsgren,
anders.forsgren@math.kth.se.
*