Michael C. Ferris

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.

The path solver: a nommonotone stabilization scheme for mixed complementarity problems

The PATH solver is an implementation of a stabilized Newton method for the solution of the Mixed Complementarity Problem. The stabilization scheme employs a path-generation procedure which is used to construct a piecewise-linear path from the current point to the Newton point; a step length acceptance criterion and a non-monotone pathsearch are then used to choose the next iterate. The algorithm is shown to be globally convergent under assumptions which generalize those required to obtain similar results in the smooth case. Several impleέentation issues are discussed, and extensive computational results obtained from problems commonly found in the literature are given

Complementarity problems in GAMS and the PATH solver

Abstract A fundamental mathematical problem is to find a solution to a square system of nonlinear equations. There are many methods to approach this problem, the most famous of which is Newtons method. In this paper, we describe a generalization of this problem, the complementarity problem. We show how such problems are modeled within the GAMS modeling language and provide details about the PATH solver, a generalization of Newtons method, for finding a solution. While the modeling format is applicable in many disciplines, we draw the examples in this paper from an economic background. Finally, some extensions of the modeling format and the solver are described.

Optimal Transmission Switching

In this paper, we formulate the problem of finding an optimal generation dispatch and transmission topology to meet a specific inflexible load as a mixed integer program. Our model is a mixed-integer linear program because it employs binary variables to represent the state of the equipment and linear relationships to describe the physical system. We find that on the standard 118-bus IEEE test case a savings of 25% in system dispatch cost can be achieved.

Co-Optimization of Generation Unit Commitment and Transmission Switching With N-1 Reliability

Currently, there is a national push for a smarter electric grid, one that is more controllable and flexible. The full control of transmission assets are not currently built into electric network optimization models. Optimal transmission switching is a straightforward way to leverage grid controllability: to make better use of the existing system and meet growing demand with existing infrastructure. Previous papers have shown that optimizing the network topology improves the dispatch of electrical networks. Such optimal topology dispatch can be categorized as a smart grid application where there is a co-optimization of both generators and transmission topology. In this paper we present a co-optimization formulation of the generation unit commitment and transmission switching problem while ensuring N-1 reliability. We show that the optimal topology of the network can vary from hour to hour. We also show that optimizing the topology can change the optimal unit commitment schedule. This problem is large and computationally complex even for medium sized systems. We present decomposition and computational approaches to solving this problem. Results are presented for the IEEE RTS 96 test case.

Weak sharp minima in mathematical programming

The notion of a sharp, or strongly unique, minimum is extended to include the possibility of a nonunique solution set. These minima will be called weak sharp minima. Conditions necessary for the solution set of a minimization problem to be a set of weak sharp minima are developed in both the unconstrained and constrained cases. These conditions are also shown to be sufficient under the appropriate convexity hypotheses. The existence of weak sharp minima is characterized in the cases of linear and quadratic convex programming and for the linear complementarity problem. In particular, a result of Mangasarian and Meyer is reproduced that shows that the solution set of a linear program is always a set of weak sharp minima whenever it is nonempty. Consequences for the convergence theory of algorithms are also examined, especially conditions yielding finite termination.

Mcplib: a collection of nonlinear mixed complementarity problems

The origins and some motivational details of a collection of nonlinear mixed complementarity problems are given. This collection serves two purposes. Firstly, it gives a uniform basis for testing currently available and new algorithms for mixed complementarity problems. Function and Jacobian evaluations for the resulting problems are provided via a GAMS interface, making thorough testing of algorithms on practical complementarity problems possible. Secondly, it gives examples of how to formulate many popular problem formats as mixed complementarity problems and how to describe the resulting problems in GAMS format. We demonstrate the ease and power of formulating practical models in the MCP format. Given these examples, it is hoped that this collection will grow to include many problems that test complementarity algorithms more fully. The collection is available by anonymous ftp. Computational results using the PATH solver covering all of these problems are described

Interior-Point Methods for Massive Support Vector Machines

We investigate the use of interior-point methods for solving quadratic programming problems with a small number of linear constraints, where the quadratic term consists of a low-rank update to a positive semidefinite matrix. Several formulations of the support vector machine fit into this category. An interesting feature of these particular problems is the volume of data, which can lead to quadratic programs with between 10 and 100 million variables and, if written explicitly, a dense Q matrix. Our code is based on OOQP, an object-oriented interior-point code, with the linear algebra specialized for the support vector machine application. For the targeted massive problems, all of the data is stored out of core and we overlap computation and input/output to reduce overhead. Results are reported for several linear support vector machine formulations demonstrating that the method is reliable and scalable.

Limit analysis of frictional block assemblies as a mathematical program with complementarity constraints

The computation of the collapse loads of discrete rigid block systems, characterized by frictional (nonassociative) and tensionless contact interfaces, is formulated and solved as a special constrained optimization problem known as a Mathematical Program with Equilibrium Constraints (MPEC). In the present instance, some of the essential constraints are defined by a complementarity system involving the orthogonality of two sign-constrained vectors. Due to its intrinsic complexity, MPECs are computationally very hard to solve. In this paper, we investigate a simple numerical scheme, involving appropriate relaxation of the complementarity term, to solve this nonstandard limit analysis problem. Some computational results are presented to illustrate potentialities of the method.

Interfaces to PATH 3.0: Design, Implementation and Usage

Several new interfaces have recently been developed requiring PATH to solve a mixed complementarity problem. To overcome the necessity of maintaining a different version of PATH for each interface, the code was reorganized using object-oriented design techniques. At the same time, robustness issues were considered and enhancements made to the algorithm. In this paper, we document the external interfaces to the PATH code and describe some of the new utilities using PATH. We then discuss the enhancements made and compare the results obtained from PATH 2.9 to the new version.