Abstract:

Complementarity problems (CPs) and variational inequalities (VIs) are prototypical equilibrium problems (EPs), as indeed are systems of equations with as many unknowns as equations. EPs arise not only as stationary conditions of optimisation problems, e.g. Lagrangian and, more generally, Karush-Kuhn-Tucker systems in nonlinear programming, but in situations for which there is no overarching optimisation model, such as games in economics and structural analysis in mechanics. If your basic model is a CP or VI, you shortly arrive at bilevel optimisation problems in which you try to estimate the parameters of your model; this is the extension of the usual nonlinear least squares format that is so familiar in engineering. These bilevel problems are called MPECs, Mathematical Programs with Equilibrium Constraints. From the theoretical and computational point of view, piecewise smooth reformulations of EPs have provided a rich path for development of that makes direct extension of classical templates such as the implicit function theorem and Newton's method. The excitement in current models and research looks beyond EPs to MPECs and bilevel games, EPECs, the latter for example in deregulated electricity markets where network effects may be significant.