Jong-Shi Pang

The linear complementarity problem

An article holding receptacle, such as an expandable envelope, is releasably secured to the steering column of an automotive vehicle by means of an elasticized band wrapped partly around the steering column hooked at both ends to a clip from which the envelope is releasably attached. The length of the band is such that, when engaged at its ends with the clip, it is under sufficient tension to be retained snugly against the steering column. The envelope is secured to the clip by releasable means such as a paper fastener and includes means to prevent skewing of the envelope about the fastener.

Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications

Over the past decade, the field of finite-dimensional variational inequality and complementarity problems has seen a rapid development in its theory of existence, uniqueness and sensitivity of solution(s), in the theory of algorithms, and in the application of these techniques to transportation planning, regional science, socio-economic analysis, energy modeling, and game theory. This paper provides a state-of-the-art review of these developments as well as a summary of some open research topics in this growing field.

Engineering and Economic Applications of Complementarity Problems

This paper gives an extensive documentation of applications of finite-dimensional nonlinear complementarity problems in engineering and equilibrium modeling. For most applications, we describe the problem briefly, state the defining equations of the model, and give functional expressions for the complementarity formulations. The goal of this documentation is threefold: (i) to summarize the essential applications of the nonlinear complementarity problem known to date, (ii) to provide a basis for the continued research on the nonlinear complementarity problem, and (iii) to supply a broad collection of realistic complementarity problems for use in algorithmic experimentation and other studies.

Strategic gaming analysis for electric power systems: an MPEC approach

Transmission constraints and market concentration may prevent power markets from being fully competitive, allowing firms to exercise market power and raise prices above marginal cost. We present a strategic gaming model for analyzing such markets; it represents an oligopolistic market economy consisting of several dominant firms in an electric power network. Each generating firm submits bids to an ISO, choosing its bids to maximize profits subject to anticipated reactions by rival firms. The single-firm model is formulated as a mathematical program with equilibrium constraints (MPEC) with a parameter-dependent spatial price equilibrium problem as the inner problem. Power flows and pricing strategies are constrained by the ISOs linearized DC optimal power flow (OFF) model. A penalty interior point algorithm is used to compute a local optimal solution of the MPEC. Numerical examples based on a 30 bus network are presented, including multi-firm Nash equilibria in which each player solves an MPEC of the single-firm type.

Oligopolistic competition in power networks: a conjectured supply function approach

Conjectured supply function (CSF) models of competition among power generators on a linearized DC network are presented. As a detailed survey of the power market modeling literature shows, CSF models differ from previous approaches in that they represent each generation companys (GenCo) conjectures regarding how rival firms will adjust sales in response to price changes. The CSF approach is a more realistic and flexible framework for modeling imperfect competition than other models for three reasons. First, the models include as a special case the Cournot conjecture that rivals will not change production if prices change; thus, the CSF framework is more general. Second, Cournot models cannot be used when price elasticity of demand is zero, but the proposed models can. Third, unlike supply function equilibrium models, CSF equilibria can be calculated for large transmission networks. Existence and uniqueness properties for prices and profits are reported. An application shows how transmission limits and strategic interactions affect equilibrium prices under forced divestment of generation.

Error bounds in mathematical programming

Originated from the practical implementation and numerical considerations of iterative methods for solving mathematical programs, the study of error bounds has grown and proliferated in many interesting areas within mathematical programming. This paper gives a comprehensive, state-of-the-art survey of the extensive theory and rich applications of error bounds for inequality and optimization systems and solution sets of equilibrium problems.

Nonsmooth Equations: Motivation and Algorithms

This paper reports on some recent developments in the area of solving of nonsmooth equations by generalized Newton methods. The emphasis is on three topics: motivation, characterization of superlin...

Iterative methods for variational and complementarity problems

In this paper, we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems. Included among such methods are the Newton and several successive overrelaxation algorithms. For the most part, the study is concerned with the family of linear approximation methods. These are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems. Convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches. The paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria. Finally, several convergence results are obtained for some nonlinear approximation methods.

Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games

In Pang and Fukushima (Comput Manage Sci 2:21–56, 2005), a sequential penalty approach was presented for a quasi-variational inequality (QVI) with particular application to the generalized Nash game. To test the computational performance of the penalty method, numerical results were reported with an example from a multi-leader-follower game in an electric power market. However, due to an inverted sign in the penalty term in the example and some missing terms in the derivatives of the firms’ Lagrangian functions, the reported numerical results in Pang and Fukushima (Comput Manage Sci 2:21–56, 2005) are incorrect. Since the numerical examples of this kind are scarce in the literature and this particular example may be useful in the future research, we report the corrected results.

The Generalized Quasi-Variational Inequality Problem

In this paper, we introduce the generalized quasi-variational inequality problem and develop a theory for the existence of solution. This new problem includes as special cases two existing generalizations of the classical variational inequality problem. Relationship with a certain implicit complementarity problem is also studied.