equilibrium problems with equilibrium constraints - Convex ...

Abstract
Equilibrium problem with equilibrium constraints (EPECs) often arise in engineering
and economics applications. One important application of EPECs is the
multi-leader-follower game in economics, where each leader is solving a Stackelberg
game formulated as a mathematical program with equilibrium constraints
(MPEC). Motivated by applied EPEC models for studying the strategic behavior
of generating firms in deregulated electricity markets, the aim of this thesis is to
study theory, algorithms, and new applications for EPECs.
We begin by reviewing the stationarity conditions and algorithms for MPECs.
Then, we generalize Scholtes’s regularization scheme for solving MPECs. We define
EPEC stationarity concepts in Chapter 3. We propose a sequential nonlinear
complementarity (SNCP) method for solving EPECs and establish its convergence.
We present the numerical results of the SNCP method and give a comparison
with two best-reply iterations, nonlinear Jacobi and nonlinear Gauss-Seidel,
on a set of randomly generated test problems. The computational experience to
date shows that both the SNCP algorithm and the nonlinear Gauss-Seidel method
outperform the nonlinear Jacobi method.
We investigate the issue of existence of an EPEC solution in Chapter 4. In
general, an EPEC solution may not exist because of nonconvexity of the associated
MPECs. However, we show that the existence result can be established for
the spot-forward market model proposed by Allaz and Vila and the two-period
Cournot game studied by Saloner. We observe that the mathematical structure
of the spot-forward market model is similar to that of the multiple leader Stackelberg
model analyzed by Sherali when new variables are introduced for spot
market sales. Consequently, we are able to adapt Sherali’s analysis to establish
the existence of a forward market equilibrium for M asymmetric producers with
nonidentical linear cost functions.
In Chapter 5, we present a novel MPEC approach for computing solutions of
incentive problems in economics. Specifically, we consider deterministic contracts
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