Equilibrium modeling and solution approaches inspired by nonconvex bilevel programming

Abstract

This paper introduces the concept of optimization equilibrium as an equivalently versatile definition of a generalized Nash equilibrium for multi-agent non-cooperative games. Through this modified definition of equilibrium, we draw precise connections between generalized Nash equilibria, feasibility for bilevel programming, the Nikaido-Isoda function, and classic arguments involving Lagrangian duality and social welfare maximization. Significantly, this is all in a general setting without the assumption of convexity. Along the way, we introduce the idea of minimum disequilibrium as a solution concept that reduces to traditional equilibrium when equilibrium exists. The connections with bilevel programming and related semi-infinite programming permit us to adapt global optimization methods for those classes of problems, such as constraint generation or cutting plane methods, to the problem of finding a minimum disequilibrium solution. We show that this method works, both theoretically and with a numerical example, even when the agents are modeled by mixed-integer programs.

Pooling problems under perfect and imperfect competition

“…As will become clear, we prefer to include π to allow for external constraints and more complex interactions between the players. A proof of the equivalence between these two representations is given in [18].…”

“…Equilibrium problems can also be modeled as bilevel optimization problems. For this work, we adapt the approach of Harwood et al [18], which introduces an exact method for equilibrium problems with nonconvex structures. Meanwhile, two commonly used bilevel models are: (1) Mathematical programs with equilibrium constraints (MPECs), which are optimization problems whose constraints include equilibrium conditions (e.g., KKT conditions of a lower-level optimization problem or Nash-Cournot game).…”

“…One is often interested in maximizing social welfare, the sum of producer and consumer surplus (i.e., the sum of the objective functions in ( 5) and ( 6)), to determine static long-run general equilibrium conditions of a perfectly competitive game. Moreover, solving for the maximum social welfare yields an equilibrium, if one exists, even if the model is nonconvex [18]. Under the aforementioned assumptions and after requiring the quantity consumed q n of output stream n ∈ N out to equal the quantity produced j∈J f out jn , the maximum social welfare problem becomes a bilinearly-constrained optimization problem with a concave quadratic objective function (an MIQCQP):…”

“…This reformulation has been considered in [4,26,30,31] and we recognize it as a semi-infinite program. This class of problems and their numerical solution methods informs the approach from [18] for finding equilibrium. In this work, we are particularly interested in the special case when the upper-level objective function φ(π, x) is trivial and everywhere equal to zero, in which case (BLP-R) reduces to a feasibility problem, albeit a challenging one given the presence of "infinite" constraints (11).…”

“…In this work, we are particularly interested in the special case when the upper-level objective function φ(π, x) is trivial and everywhere equal to zero, in which case (BLP-R) reduces to a feasibility problem, albeit a challenging one given the presence of "infinite" constraints (11). Harwood et al [18] offers an attractive alternative to this feasibility problem by replacing the trivial objective with one that seeks to minimize the violation of the infinite constraint (11). While there are many options for measuring the violation, or in our context the "amount of disequilibrium," we use arguably the simplest choice of minimizing the absolute (L1) distance between p * (π) and p(x, π), leading to the following model:…”