Nonlinear programming advances in mathematical programming with complementarity constraints

Given a suitably parameterised family of equilibrium models and a higher level criterion by which to measure an equilibrium state, mathematical programs with equilibrium constraints (MPECs) provide a framework for selecting good or even optimal parameters. An example is toll design in traffic networks, which attempts to reduce total travel time by choosing which arcs to toll and what toll levels to impose. Here, a Wardrop equilibrium describes the traffic response to each toll design. Communication networks also have a deep literature on equilibrium flows that suggest some MPECs. We focus on mathematical programs with complementarity constraints (MPCCs), a subclass of MPECs for which the lower level equilibrium system can be formulated as a complementarity problem. An MPCC can be immediately written as a nonlinear program (NLP) but, regrettably, the constraints of the latter lack some stability properties that are considered essential in analysing and solving NLPs. A related issue is that MPECs and MPCCs are generally nonconvex, which is a consequence of the upper level objective clashing with users’ objectives in the lower level equilibrium program. While global optimisation of MPECs and MPCCs is in a state of infancy, the last decade of research has paved the way for finding local solutions of MPCCs via standard NLP techniques if not off-theshelf NLP software. We give a selective review of these advances. Special attention is paid to Lagrangian techniques which have become essential in describing what “stationary” means for MPCCs, and how to solve MPCCs computationally.