Todd S. Munson

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.

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.

Addressing failures in exascale computing

We present here a report produced by a workshop on ‘Addressing failures in exascale computing’ held in Park City, Utah, 4–11 August 2012. The charter of this workshop was to establish a common taxonomy about resilience across all the levels in a computing system, discuss existing knowledge on resilience across the various hardware and software layers of an exascale system, and build on those results, examining potential solutions from both a hardware and software perspective and focusing on a combined approach. The workshop brought together participants with expertise in applications, system software, and hardware; they came from industry, government, and academia, and their interests ranged from theory to implementation. The combination allowed broad and comprehensive discussions and led to this document, which summarizes and builds on those discussions.

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.

Solving multi-leader–common-follower games

Multi-leader–common-follower games arise when modelling two or more competitive firms, the leaders, that commit to their decisions prior to another group of competitive firms, the followers, that react to the decisions made by the leaders. These problems lead in a natural way to equilibrium problems with equilibrium constraints (EPECs). We develop a characterization of the solution sets for these problems and examine a variety of nonlinear optimization and nonlinear complementarity formulations of EPECs. We distinguish two broad cases: problems where the leaders can cost-differentiate and problems with price-consistent followers. We demonstrate the practical viability of our approach by solving a range of medium-sized test problems.

Feasible descent algorithms for mixed complementarity problems

Abstract.In this paper we consider a general algorithmic framework for solving nonlinear mixed complementarity problems. The main features of this framework are: (a) it is well-defined for an arbitrary mixed complementarity problem, (b) it generates only feasible iterates, (c) it has a strong global convergence theory, and (d) it is locally fast convergent under standard regularity assumptions. This framework is applied to the PATH solver in order to show viability of the approach. Numerical results for an appropriate modification of the PATH solver indicate that this framework leads to substantial computational improvements.

Leader-Follower Equilibria for Electric Power and NO x Allowances Markets

This paper investigates the ability of the largest producer in an electricity market to manipulate both the electricity and emission allowances markets to its advantage. A Stackelberg game to analyze this situation is constructed in which the largest firm plays the role of the leader, while the medium-sized firms are treated as Cournot followers with price-taking fringes that behave competitively in both markets. Since there is no explicit representation of the best-reply function for each follower, this Stackelberg game is formulated as a large-scale mathematical program with equilibrium constraints. The best-reply functions are implicitly represented by a set of nonlinear complementarity conditions. Analysis of the computed solution for the Pennsylvania–New Jersey–Maryland electricity market shows that the leader can gain substantial profits by withholding allowances and driving up NOx allowance costs for rival producers. The allowances price is higher than the corresponding price in the Nash–Cournot case, although the electricity prices are essentially the same.

Semismooth support vector machines

Abstract.Support vector machines can be posed as quadratic programming problems in a variety of ways. This paper investigates a formulation using the two-norm for the misclassification error that leads to a positive definite quadratic program with a single equality constraint under a duality construction. The quadratic term is a small rank update to a diagonal matrix with positive entries. The optimality conditions of the quadratic program are reformulated as a semismooth system of equations using the Fischer-Burmeister function and a damped Newton method is applied to solve the resulting problem. The algorithm is shown to converge from any starting point with a Q-quadratic rate of convergence. At each iteration, the Sherman-Morrison-Woodbury update formula is used to solve the key linear system. Results for a large problem with 60 million observations are presented demonstrating the scalability of the proposed method on a personal computer. Significant computational savings are realized as the inactive variables are identified and exploited during the solution process. Further results on a small problem separated by a nonlinear surface are given showing the gains in performance that can be made from restarting the algorithm as the data evolves.

Mesh shape-quality optimization using the inverse mean-ratio metric

Meshes containing elements with bad quality can result in poorly conditioned systems of equations that must be solved when using a discretization method, such as the finite-element method, for solving a partial differential equation. Moreover, such meshes can lead to poor accuracy in the approximate solution computed. In this paper, we present a nonlinear fractional program that relocates the vertex coordinates of a given mesh to optimize the average element shape quality as measured by the inverse mean-ratio metric. To solve the resulting large-scale optimization problems, we apply an efficient implementation of an inexact Newton algorithm that uses the conjugate gradient method with a block Jacobi preconditioner to compute the direction. We show that the block Jacobi preconditioner is positive definite by proving a general theorem concerning the convexity of fractional functions, applying this result to components of the inverse mean-ratio metric, and showing that each block in the preconditioner is invertible. Numerical results obtained with this special-purpose code on several test meshes are presented and used to quantify the impact on solution time and memory requirements of using a modeling language and general-purpose algorithm to solve these problems.

A comparison of two optimization methods for mesh quality improvement

We compare inexact Newton and block coordinate descent optimization methods for improving the quality of a mesh by repositioning the vertices, where the overall quality is measured by the harmonic mean of the mean-ratio metric. The effects of problem size, element size heterogeneity, and various vertex displacement schemes on the performance of these algorithms are assessed for a series of tetrahedral meshes.