Regularity, stability, and computation of equilibria

The notion of equilibrium plays a prominent role in science and engineering. Classically, equilibria are described by systems of equations, but if constraints are involved in the model, the equilibrium relations may take the form of a variational inequality (VI). Optimality conditions for optimization problems of different kinds, varying from mathematical programming to calculus of variations and optimal control, are standardly described by VIs. VIs may also arise in modeling of equilibrium processes in physics and economics. Examples are sweeping processes in continuum mechanics and Walrasian equilibria of product markets. An important and desirable property of equilibria is their stability, that is, the property that an equilibrum does not disappear or change abruptly as a result of small changes in the model. This property is also often necessary for efficient computations of equilibria. The concept of metric regularity has emerged in the past 40 years (having its roots in classical works of Banach, Lyusternik, Graves, and others) as a powerful tool for investigation of equilibrium stability. This concept will be systematically employed in the project for VIs, parametric VIs, and differential VIs, describing equilibria in three different classes of models: static systems, exogenously changing systems, and endogenously evolving systems, respectively. Extended versions of a Walrasian model for economic equilibria will be used as workbench examples for the above three classes of VIs. The project consists of three main parts: Metric regularity and conditioning, where the goal is to develop theoretical and numerical tools for estimation of the radius of metric regularity of classes of VIs, and for the workbench examples in particular. The radius of metric regularity gives information of how much a model can be disturbed before it experiences an abrupt change of its stability. Parametric equilibria and path-following, where the goal is to develop predictor-corrector continuation methods for parametric VIs. These may ensure high-order approximations despite of the intrinsically non-smooth character of the underlying equilibrium problem. The direct application of predictor-corrector continuation methods will be compared, and possibly combined, with the semi-smooth (quasi-) Newton method applied to the reformulation of the VIs as non-smooth equations. Differential variational inequalities and sweeping processes, where the development of high order numerical approximation schemes is the main goal. These will be implemented in new methods for solving optimal control problems, computing Nash equilibria in differential games, and computing dynamic economic equilibria. The ultimate goal of the project is to achieve a better understanding of a broad class of VIs and stability of their equilibria, and to develop efficient numerical schemes for solving large-scale equilibrium problems. As a primal application we envisage economic equilibria, but applications to problems of optimal control or differential games that have a broader range of applications are also targeted.

The notion of equilibrium plays a prominent role in science and engineering. Classically, equilibria are described by systems of equations, but if constraints are involved in the model, the equilibrium relations may take the form of the so-called variational inequalities (VIs). Optimality conditions for optimization problems of different kinds, varying from mathematical programming to calculus of variations, optimal control and dynamic games, are standardly described by VIs. Such also arise in modelling of equilibrium processes in physics and economics. The project developed new tools for computing equilibria of various kinds. Dominantly, these are algorithms with fast convergence, so that they can be utilized for optimal control of fast processes in engineering, especially such arising in power electronics. Strongly related with the analysis of such algorithms is the question of how the equilibria change as a result of changes in the model parameters. A typical economic example is the dependence of the market (equilibrium) prices of certain goods on given exogenous factors. Thus the equilibria may have their own dynamics and computing the evolution, and even more, controlling the evolution, of equilibria can be a hard task which requires specific mathematical tools. The project developed several new theoretical and computational tools for investigation of variational inequalities and the dependence of their solutions on data, especially such for optimal control of dynamic processes.