equilibrium problems with equilibrium constraints - Convex ...

Chapter 1
Introduction
An equilibrium problem with equilibrium constraints (EPEC) is a member of a new
class of mathematical programs that often arise in engineering and economics
applications. One important application of EPECs is the multi-leader-follower
game [47] in economics. In noncooperative game theory, the well-known Stackelberg
game (single-leader-multi-follower game) can be formulated as an optimization
problem called a mathematical program with equilibrium constraints (MPEC)
[33, 43], in which followers’ optimal strategies are solutions of complementarity
problems or variational inequality problems based on the leader’s strategies.
Analogously, the more general problem of finding equilibrium solutions of a multileader-follower
game, where each leader is solving a Stackelberg game, is formulated
as an EPEC. Consequently, one may treat an EPEC as a two-level hierarchical
problem, which involves finding equilibria at both lower and upper levels.
More generally, an EPEC is a mathematical program to find equilibria that simultaneously
solve several MPECs, each of which is parameterized by decision
variables of other MPECs.
Motivated by applied EPEC models for studying the strategic behavior of
generating firms in deregulated electricity markets [5, 22, 49], the aim of this
thesis is to study the stationarities, algorithms, and new applications for EPECs.
The main contributions of this thesis are summarized below.
In Chapter 2, we review the stationarity conditions and algorithms for mathematical
programs with equilibrium constraints. We generalize Scholtes’s regularization
scheme for solving MPECs and establish its convergence. Chapter 3
begins by defining EPEC stationarities, followed by a new class of algorithm,
called a sequential nonlinear complementarity (SNCP) method, which we propose
for solving EPECs. Numerical approaches used by researchers in engineering and
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