Equilibrium problems with equilibrium constraints: Stationarities, algorithms, and applications

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 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 defining EPEC stationarity concepts. We then 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 or 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 solving the static moral-hazard problem in economics. We consider deterministic contracts as well as contracts with action and/or compensation lotteries and formulate each case as an MPEC. We propose a hybrid procedure that combines the best features of the MPEC approach and the LP lottery approach. Numerical results on an example show that the hybrid procedure outperforms the LP lottery approach in both computational time and solution quality in terms of the optimal objective value.