Mathematical Program Networks

TL;DR

Mathematical Program Networks (MPNs) unify various complex interconnected decision problems into a single framework, enabling consistent equilibrium analysis and solution algorithms across diverse applications.

Contribution

This work introduces MPNs as a new framework to model and analyze interconnected mathematical programs with a unified approach to equilibrium computation.

Findings

MPNs encompass Nash Equilibrium, multilevel optimization, and EPECs.

A common solution approach applies across different problem types.

Framework enhances modeling flexibility and solution analysis.

Abstract

Mathematical Program Networks (MPNs) are introduced in this work. An MPN is a collection of interdependent Mathematical Programs (MPs) which are to be solved simultaneously, while respecting the connectivity pattern of the network defining their relationships. The network structure of an MPN impacts which decision variables each constituent mathematical program can influence, either directly or indirectly via solution graph constraints representing optimal decisions for their decedents. Many existing problem formulations can be formulated as MPNs, including Nash Equilibrium problems, multilevel optimization problems, and Equilibrium Programs with Equilibrium Constraints (EPECs), among others. The equilibrium points of an MPN correspond with the equilibrium points or solutions of these other problems. By thinking of a collection of decision problems as an MPN, a common definition of equilibrium can be used regardless of relationship between problems, and the same algorithms can be used to compute solutions. The presented framework facilitates modeling flexibility and analysis of various equilibrium points in problems involving multiple mathematical programs.