MODLER: Modeling by object-driven linear elemental relations

This paper describes a system to represent linear programming models and their instances. In addition to a modeling language, MODLER has an extensive query capability which includes a multi-view architecture. Further, randomization options provide rapid prototyping. The MODLER system is part of a workbench for building and managing decision support systems that are based on linear programming.This research was supported by a consortium of industries: Amoco Oil Company, Shell Development Company, Chesapeake Decision Science, GAMS Development Corp., Ketron Management Science, MathPro, Inc., Optimal Methods, Inc., and XMP Software, Inc. Partial support was also provided by the Office of Naval Research (Contract No. N-00014-88-K-0104).     

LP modeling languages for personal computers: A comparison

The purpose of this paper is to compare and contrast the modeling capabilities of seven algebraic modeling languages (ML) available today, namely, AMPL, GAMS, LINGO, LPL, MPL, PC-PROG and XPRESS-LP. In general, these MLs do an excellent job of providing an interface with which the modeler can specify an algebraically formatted linear program (LP). That is, each ML provides a substantial improvement in time and convenience over the matrix generator/report writers of the last few decades. Further, each of the MLs provides: (1) significant flexibility in model specification, instantiation and modification, (2) effective and efficient conversion from algebraic to solver format, and (3) an understandable and, for the most part, self-documenting model representation. In addition, each of the MLs is constantly being updated and upgraded to provide additional capabilities sought by practitioners and users. However, as shown in the fifteen tables provided in the body of this paper, each ML has its own set of competitive advantages. For example, the most integrated environments (i.e. those integrating the modeling language with a full-screen editor, data import capabilities and a solver) are provided by LINGO and PC-PROG. The most user-friendly interfaces are provided by MPL and PC-PROG, both of which provide window-based interfaces to create models and pop-up windows to display error messages; MPL also uses pull-down menus to specify various operations, whereas PC-PROG uses function keys for operational control. Package costs are led by a current (March, 1991) introductory offer from LINGO. Modeling effectiveness, especially with respect to flexibility in specifying arithmetic statements, is led by GAMS and LPL. Model compactness, as measured by the number of lines required to specify a model, is led by AMPL, LPL, MPL and PC-PROG; LPL, MPL and PC-PROG also provide context sensitive editors which automatically position the cursor where the error was detected. And finally, the most comprehensive user documentation is provided by GAMS, whereas GAMS, LINGO and LPL provide extensive libraries of sample models for those users who learn by example.     

Scenario formulation in an algebraic modelling language

Algebraic modelling languages have simplified management of many types of large linear programs but have not specifically supported stochastic modelling. This paper considers modelling language support for multistage stochastic linear recourse problems with finite distributions. We describe basic language requirements for formulation of finite event trees in algebraic modelling languages and show representative problems in AMPL using three commonly used scenario types.     

Solving Bilevel Linear Programs Using Multiple Objective Linear Programming

We present an algorithm for solving bilevel linear programs that uses simplex pivots on an expanded tableau. The algorithm uses the relationship between multiple objective linear programs and bilevel linear programs along with results for minimizing a linear objective over the efficient set for a multiple objective problem. Results in multiple objective programming needed are presented. We report computational experience demonstrating that this approach is more effective than a standard branch-and-bound algorithm when the number of leader variables is small.     

OMNI Model Management System

OMNI is a model management system (MMS). Its purpose is to provide the software tool to build, maintain and make use of practical application systems. These applications commonly involve Linear Programming and its extensions (LP). OMNI MMS includes a powerful database, many calculation and logic capabilities, report writing functions, graphics and connectivity to outside systems. Because of these, it also has other uses such as for simulation, scheduling and data management.     

Modeling languages and Condor: metacomputing for optimization

A generic framework is postulated for utilizing the computational resources provided by a metacomputer to concurrently solve a large number of optimization problems generated by a modeling language. An example of the framework using the Condor resource manager and the AMPL and GAMS modeling languages is provided. A mixed integer programming formulation of a feature selection problem from machine learning is used to test the mechanism developed. Due to this application’s computational requirements, the ability to perform optimizations in parallel is necessary in order to obtain results within a reasonable amount of time. Details about the simple and easy to use tool and implementation are presented so that other modelers with applications generating many independent mathematical programs can take advantage of it to significantly reduce solution times. Received: October 28, 1998 / Accepted: December 01, 1999?Published online June 8, 2000     

Constraint propagation,relational arithmetic in AI systems and mathematical programs

This paper explores the interrelationships between methods developed in mathematical programming to discover the structure of constraint (feasibility) sets and constraint propagation over networks used by some AI systems to perform inferences about quantities. It is shown that some constraint set problems in mathematical programming are equivalent to inferencing problems for constraint networks with interval labels. This makes the inference and query capabilities associated with AI systems that use logic programming, directly accessible to mathematical programming systems. On the other hand, traditional and newer methods which mathematical programming uses to obtain information about its associated feasibility set can be used to determine the propagation of constraints in a network of nodes of an AI system. When viewed from this point of view, AI problems can access additional mathematical programming analytical tools including new ways to incorporate qualitative data into constraint sets via interval and fuzzy arithmetic.This work was partially supported by the Industrial Consortium to Develop an Intelligent Mathematical Programming System — Amoco Oil Company, General Research Corporation, Ketron Management Science, Shell Oil Company, MathPro, and US West Advanced Technologies.     

On the formulation of stochastic linear programs using algebraic modelling languages

This paper considers extensions to algebraic modelling languages to support formulation, instantiation and solver integration for stochastic linear programs (SLPs). We present a taxonomy of SLP problem types and analyze formulation requirements including distribution handling by class of problem. We demonstrate suggested formulations for most problem classes, show solver input in the S-MPS standard, and propose consistency checks for constraints involving stochastic data items. Some unresolved difficulties are identified.     

OSiL: An instance language for optimization

Distributed computing technologies such as Web Services are growing rapidly in importance in today’s computing environment. In the area of mathematical optimization, it is common to separate modeling languages from optimization solvers. In a completely distributed environment, the modeling language software, solver software, and data used to generate a model instance might reside on different machines using different operating systems. Such a distributed environment makes it critical to have an open standard for exchanging model instances. In this paper we present OSiL (Optimization Services instance Language), an XML-based computer language for representing instances of large-scale optimization problems including linear programs, mixed-integer programs, quadratic programs, and very general nonlinear programs. OSiL has two key features that make it much superior to current standard forms for optimization problem instances. First, it uses the object-oriented features of XML schemas to efficiently represent nonlinear expressions. Second, its XML schema maps directly into a corresponding in-memory representation of a problem instance. The in-memory representation provides a robust application program interface for general nonlinear programming, facilitates reading and writing postfix, prefix, and infix formats to and from the nonlinear expression tree, and makes the expression tree readily available for function and derivative evaluations.     

Efficient Management of Multiple Sets to Extract Complex Structures from Mathematical Programs

Most of the applied models written with an algebraic modeling language involve simultaneously several dimensions such as materials, location, time or uncertainty. The information about dimensions available in the algebraic formulation is usually sufficient to retrieve different block structures from mathematical programs. These structured problems can then be solved by adequate solution techniques. To illustrate this idea we focus on stochastic programming problems with recourse. Taking into account both time and uncertainty dimensions of these problems, we are able to retrieve different customized structures in their constraint matrices. We applied the Structure Exploiting Tool to retrieve the structure from models built with the GAMS modeling language. The underlying mathematical programs are solved with the decomposition algorithm that applies interior point methods. The optimization algorithm is run in a sequential and in a parallel computing environment.     

Implementation aspects of interactive multiobjective optimization for modeling environments: the case of GAMS-NIMBUS

Interactive multiobjective optimization methods have provided promising results in the literature but still their implementations are rare. Here we introduce a core structure of interactive methods to enable their convenient implementation. We also demonstrate how this core structure can be applied when implementing an interactive method using a modeling environment. Many modeling environments contain tools for single objective optimization but not for interactive multiobjective optimization. Furthermore, as a concrete example, we present GAMS-NIMBUS Tool which is an implementation of the classification-based NIMBUS method for the GAMS modeling environment. So far, interactive methods have not been available in the GAMS environment, but with the GAMS-NIMBUS Tool we open up the possibility of solving multiobjective optimization problems modeled in the GAMS modeling environment. Finally, we give some examples of the benefits of applying an interactive method by using the GAMS-NIMBUS Tool for solving multiobjective optimization problems modeled in the GAMS environment.     

Establishing strict dominance between alternatives with special type of incomplete information

A special model for establishing dominance with decision maker's incomplete information is proposed in multi-attribute decision problem under certainty. Incomplete information is a set of linear inequalities about utilities as well as attribute weights. Since a general model presented in a prior work uses a linear programming technique, a number of linear programs should be solved for checking dominance. With the proposed method, we can check strict dominance between alternatives by simple matrix operation without solving linear programs, thus reducing computational time.     

A new algorithm for solving the general quadratic programming problem

For the general quadratic programming problem (including an equivalent form of the linear complementarity problem) a new solution method of branch and bound type is proposed. The branching procedure uses a well-known simplicial subdivision and the bound estimation is performed by solving certain linear programs.     

Restorations of punctured languages and similarity of languages

Punctured languages are languages whose words are partial words in the sense that the letters at some positions are unknown. We investigate to which extent restoration of punctured languages is possible if the number of unknown positions or the proportion of unknown positions per word, respectively, is bounded, and we study their relationships for different boundings. The considered restoration classes coincide with similarity classes according to some kind of similarity for languages. Thus all results we can also formulate in the language of similarity. We show some hierarchies of similarity classes for each class from the Chomsky hierarchy and prove the existence of linear languages which are not δ ‐similar to any regular language for any δ < ½. For δ ≥ ½ this is unknown but it could only be possible in the case of non‐slender linear languages. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Relaxation methods for monotropic programs

We propose a dual descent method for the problem of minimizing a convex, possibly nondifferentiable, separable cost subject to linear constraints. The method has properties reminiscent of the Gauss-Seidel method in numerical analysis and uses the-complementary slackness mechanism introduced in Bertsekas, Hosein and Tseng (1987) to ensure finite convergence to near optimality. As special cases we obtain the methods in Bertsekas, Hosein and Tseng (1987) for network flow programs and the methods in Tseng and Bertsekas (1987) for linear programs.Work supported by the National Science Foundation under grant NSF-ECS-8519058 and by the Army Research Office under grant DAAL03-86-K-0171.This author is also supported by Canadian NSERC under Grant U0505.     

Nonanticipative duality,relaxations, and formulations for chance-constrained stochastic programs

We propose two new Lagrangian dual problems for chance-constrained stochastic programs based on relaxing nonanticipativity constraints. We compare the strength of the proposed dual bounds and demonstrate that they are superior to the bound obtained from the continuous relaxation of a standard mixed-integer programming (MIP) formulation. For a given dual solution, the associated Lagrangian relaxation bounds can be calculated by solving a set of single scenario subproblems and then solving a single knapsack problem. We also derive two new primal MIP formulations and demonstrate that for chance-constrained linear programs, the continuous relaxations of these formulations yield bounds equal to the proposed dual bounds. We propose a new heuristic method and two new exact algorithms based on these duals and formulations. The first exact algorithm applies to chance-constrained binary programs, and uses either of the proposed dual bounds in concert with cuts that eliminate solutions found by the subproblems. The second exact method is a branch-and-cut algorithm for solving either of the primal formulations. Our computational results indicate that the proposed dual bounds and heuristic solutions can be obtained efficiently, and the gaps between the best dual bounds and the heuristic solutions are small.     

A constraint generation algorithm for large scale linear programs using multiple-points separation

In order to solve linear programs with a large number of constraints, constraint generation techniques are often used. In these algorithms, a relaxation of the formulation containing only a subset of the constraints is first solved. Then a separation procedure is called which adds to the relaxation any inequality of the formulation that is violated by the current solution. The process is iterated until no violated inequality can be found. In this paper, we present a separation procedure that uses several points to generate violated constraints. The complexity of this separation procedure and of some related problems is studied. Also, preliminary computational results about the advantages of using multiple-points separation procedures over traditional separation procedures are given for random linear programs and survivable network design. They illustrate that, for some specific families of linear programs, multiple-points separation can be computationally effective.     

Computational Experience with a Safeguarded Barrier Algorithm for Sparse Nonlinear Programming

We describe an enhanced version of the primal-dual interior point algorithm in Lasdon, Plummer, and Yu (ORSA Journal on Computing, vol. 7, no. 3, pp. 321–332, 1995), designed to improve convergence with minimal loss of efficiency, and designed to solve large sparse nonlinear problems which may not be convex. New features include (a) a backtracking linesearch using an L1 exact penalty function, (b) ensuring that search directions are downhill for this function by increasing Lagrangian Hessian diagonal elements when necessary, (c) a quasi-Newton option, where the Lagrangian Hessian is replaced by a positive definite approximation (d) inexact solution of each barrier subproblem, in order to approach the central trajectory as the barrier parameter approaches zero, and (e) solution of the symmetric indefinite linear Newton equations using a multifrontal sparse Gaussian elimination procedure, as implemented in the MA47 subroutine from the Harwell Library (Rutherford Appleton Laboratory Report RAL-95-001, Oxfordshire, UK, Jan. 1995). Second derivatives of all problem functions are required when the true Hessian option is used. A Fortran implementation is briefly described. Computational results are presented for 34 smaller models coded in Fortran, where first and second derivatives are approximated by differencing, and for 89 larger GAMS models, where analytic first derivatives are available and finite differencing is used for second partials. The GAMS results are, to our knowledge, the first to show the performance of this promising class of algorithms on large sparse NLP's. For both small and large problems, both true Hessian and quasi- Newton options are quite reliable and converge rapidly. Using the true Hessian, INTOPT is as reliable as MINOS on the GAMS models, although not as reliable as CONOPT. Computation times are considerably longer than for the other 2 solvers. However, interior point methods should be considerably faster than they are here when analytic second derivatives are available, and algorithmic improvements and problem preprocessing should further narrow the gap.