Alexander Meeraus

MINLPLib--A Collection of Test Models for Mixed-Integer Nonlinear Programming

The paper describes a new computerized collection of test models for mixed-integer nonlinear programming. Because there is no standard format for nonlinear models, the model collection is augmented with a translation server that can transform the models from their basic GAMS format into other formats, including AMPL, BARON, LGO, LINGO, and MINOPT. The translation server can also be used to transform industrial models that contain confidential information. Such transformations allow many of these models to be distributed to the research community as highly relevant algorithmic test models.

An algebraic approach to modeling

Abstract Problems constraining the successful application of large-scale models in a strategic planning environment are examined and indequate modeling languages and software are identified as the major source of difficulties. A General Algebraic Modeling System (GAMS), in use at the World Bank, is a possible solution. This system uses a language comprehensible by both people and computers and has automated most stages in the modeling process, thus making modeling quicker, cheaper, and less demanding of specialized skills, while reducing the likelihood of errors.

Frontiers in Applied General Equilibrium Modeling: Mathematical Programs with Equilibrium Constraints: Automatic Reformulation and Solution via Constrained Optimization

Constrained optimization has been extensively used to solve many large scale deterministic problems arising in economics, including, for example, square systems of equations and nonlinear programs. A separate set of models have been generated more recently, using complementarity to model various phenomenon, particularly in general equilibria. The unifying framework of mathematical programs with equilibrium constraints (MPEC) has been postulated for problems that combine facets of optimization and complementarity. This paper briefly reviews some methods available to solve these problems and described a new suite of tools for working with MPEC models. Computational results demonstrating the potential of this tool are given that automatically construct and solve a variety of different nonlinear programming reformulations of MPEC problems. This material is based on research partially supported by the National Science Foundation Grant CCR-9972372, the Air Force Office of Scientific Research Grant F49620-01-1-0040, Microsoft Corporation and the Guggenheim Foundation.

MATRIX AUGMENTATION AND PARTITIONING IN THE UPDATING OF THE BASIS INVERSE

A compact and flexible updating procedure using matrix augmentation is developed. It is shown that the representation of the updated inverse does not grow monotonically in size, and may actually decrease during certain simplex iterations. Angular structures, such as GUB, are handled naturally within the partitioning framework, and require no modifications of the simplex method.

An extended mathematical programming framework

Abstract Extended mathematical programs are collections of functions and variables joined together using specific optimization and complementarity primitives. This paper outlines a mechanism to describe such an extended mathematical program by means of annotating the existing relationships within a model to facilitate higher level structure identification. The structures, which often involve constraints on the solution sets of other models or complementarity relationships, can be exploited by modern large scale mathematical programming algorithms for efficient solution. A specific implementation of this framework is outlined that communicates structure from the GAMS modeling system to appropriate solvers in a computationally beneficial manner. Example applications are taken from chemical engineering.

Grid-Enabled Optimization with GAMS

We describe a framework for modeling optimization problems for solution on a grid computer. The framework is easy to adapt to multiple grid engines and can seamlessly integrate evolving mechanisms from particular computing platforms. It facilitates the widely used master-worker model of computing and is shown to be flexible and powerful enough for a large variety of optimization applications. In particular, we summarize a number of new features of the GAMS modeling system that provide a lightweight, portable, and powerful framework for optimization on a grid. We provide downloadable examples of its use for embarrasingly parallel financial applications, decomposition of complementarity problems, and for solving very difficult mixed-integer programs to optimality. Computational results are provided for a number of different grid engines, including multicore machines, a pool of machines controlled by the Condor resource manager, and the grid engine from Sun Microsystems.

Computing Wardropian equilibria in a complementarity framework

This note considers alternative methods for computing Wadropian (traffic network) equilibria using a multicommodity formulation in nonlinear program and complementarity formats. These methods compute exact equilibria, they are efficient and they can be implemented with standard modeling software.

Matrix augmentation and structure preservation in linearly constrained control problems

Matrix augmentation is used for the inversion of bases associated with large linearly constrained control problems. It is shown how an efficient data structure can be maintained by keeping all state variables in the basis, and then nullifying some of them explicitly by using additional constraints. The proposed methodology, together with a basis updating scheme based on augmentation, forms the skeleton for an in-core algorithm using either the revised simplex method or the generalized reduced gradient method.

Quality Assurance and Global Optimization

GAMS Development and ARKI Consulting use Quality Assurance (QA) as an integral part of their software development and software publishing process. Research and development in the global optimization area has resulted in promising implementations. Initiated by customer demand, we have been adding three different global codes, BARON, LGO, and OQNLP, to our portfolio of nonlinear optimization solvers. As part of our QA effort towards the integration and testing of these new global solvers an open architecture for reliability and performance testing of mixed-integer nonlinear optimization codes has been released. This open testing framework has been placed in the public domain (www.gamsworld.org) to serve our customers, and researchers in general, by making reproducible tests a practical proposition. We give examples illustrating the quality assurance process for obtaining performance results for local and global nonlinear and mixed-integer nonlinear programming solvers, using the existing framework tools described in this article.

Combining QCR and CHR for convex quadratic pure 0–1 programming problems with linear constraints

The convex hull relaxation (CHR) method (Albornoz in Doctoral Dissertation, 1998, Ahlatçıoğlu in Summer paper, 2007, Ahlatçıoğlu and Guignard in OPIM Dept. Report, 2010) provides lower bounds and feasible solutions on convex 0–1 nonlinear programming problems with linear constraints. In the quadratic case, these bounds may often be improved by a preprocessing step that adds to the quadratic objective function terms that are equal to 0 for all 0–1 feasible solutions yet increase its continuous minimum. Prior to computing CHR bounds, one may use Plateau’s quadratic convex reformulation (QCR) method (2006), or one of its weaker predecessors designed for unconstrained problems, the eigenvalue method of Hammer and Rubin (RAIRO 3:67–79, 1970) or the method of Billionnet and Elloumi (Math. Program, Ser. A 109:55–68, 2007). In this paper, we first describe the CHR method, and then present the QCR reformulation methods. We present computational results for convex GQAP problems.