MPEC

Mathematical Program with Equilibrium Constraints (MPEC) is a class of optimization problems in which the decision variables are constrained by equilibrium conditions that themselves depend on those variables. MPECs arise when a decision problem interacts with an underlying system whose equilibrium must hold, such as competitive markets, traffic networks, or other strategic settings. They are a generalization of bilevel optimization and complementarity problems, and they pose unique mathematical and computational challenges.

A typical MPEC can be formulated as: minimize f(x,y) subject to g(x,y) ≤ 0, h(x,y) = 0, where the variable y must be an equilibrium of a subordinate problem given x. In practice, the equilibrium requirement is often represented by the optimality conditions (such as Karush-Kuhn-Tucker conditions) of a lower-level program, or by a variational inequality or a set of complementarity constraints. This embedding leads to a single larger problem whose feasible set encodes both the upper-level decisions and the equilibrium constraints.

Variants and solution approaches: MPECs are often reformulated as mathematical programs with complementarity constraints (MPCCs) or handled through penalty, smoothing, or regularization techniques. Decomposition and specialized algorithms—such as active-set or trust-region methods—are used to exploit problem structure. Theoretical analysis frequently centers on constraint qualifications, nonconvexity, and the potential for degeneracy.

Applications span economics, energy systems, transportation and logistics, network design, and engineering, where a system-wide equilibrium interacts with decision-making. The acronym MPEC may also refer to other contexts, but Mathematical Program with Equilibrium Constraints is the principal meaning in optimization and operations research.