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Mathematical programs with complementarity constraints in traffic and telecommunications networks.

¹Judge Business School, University of Cambridge, Cambridge, UK. d.ralph@jbs.cam.ac.uk

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences

|March 8, 2008

Summary

Mathematical programs with equilibrium constraints (MPECs) optimize equilibrium states, like traffic network tolls. Research shows mathematical programs with complementarity constraints (MPCCs) can be solved using nonlinear programming (NLP) techniques.

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Area of Science:

Optimization Theory
Operations Research
Network Science

Background:

Mathematical programs with equilibrium constraints (MPECs) offer a framework for optimizing equilibrium states.
Applications include toll design in traffic networks and equilibrium flows in communication networks.
MPECs are often non-convex due to conflicting objectives between the upper-level optimization and lower-level user equilibrium.

Purpose of the Study:

To explore mathematical programs with complementarity constraints (MPCCs), a subclass of MPECs.
To demonstrate that the equilibrium system in MPCCs can be formulated as a complementarity problem and a nonlinear program (NLP).
To highlight recent advancements in solving MPCCs using standard NLP techniques.

Main Methods:

Formulating the lower-level equilibrium system as a complementarity problem.
Representing MPCCs as nonlinear programs (NLPs).
Utilizing established NLP techniques for solving MPCCs.

Main Results:

MPCCs, a subclass of MPECs, can be reformulated as NLPs.
Standard NLP techniques are effective for finding local solutions to MPCCs.
This approach facilitates the optimization of equilibrium states in complex systems.

Conclusions:

MPCCs can be effectively solved as NLPs, overcoming non-convexity challenges.
Recent research enables the application of standard NLP solvers to optimize equilibrium models.
This framework advances the optimization of systems with equilibrium constraints, such as traffic and communication networks.