Robert Fourer | Northwestern University (original) (raw)

Papers by Robert Fourer

Research paper thumbnail of Interdependence of Methods and Representations in Design of Software for Combinatorial Optimization

be entirely general (C, Fortran), or specialized for mathematical modeling (Matlab, Mathematica, ... more be entirely general (C, Fortran), or specialized for mathematical modeling (Matlab, Mathematica, Maple), or specialized for optimization as in the case of decades-old packages such as OMNI 29] and modern C++ libraries such as ILOG Solver 53]. The drawbacks of programming to describe optimization models are well known, however so-called \matrix generation" programs are hard to debug, to maintain, and to document 1 5 ]. Our focus in this paper is on higher-level representations that allow optimization models to be described non-procedurally, in terms familiar to human modelers. The continuing success of such representations | and of modeling systems based on them | testi es to their value in optimization. We survey in particular three representations popularly applied in combinatorial optimization, algebraic modeling languages, constraint logic programming languages, and netforms, or network diagrams. We rst describe the kinds of optimization methods and systems most commonly associated with these alternatives. Each p a i r o f r e p r e s e n tations is then considered, to show h o w each has been advantageous and how its advantages have begun to in uence (or ought to in uence) the design of the other. Our current research projects are described in conjunction with several of these comparisons.

Research paper thumbnail of Constraints and AI Planning

IEEE Intelligent Systems, 2005

Research paper thumbnail of A modeling language for mathematical programming

Management Science, Jan 1, 1990

Research paper thumbnail of Modeling languages versus matrix generators for linear programming

ACM Transactions on Mathematical Software (TOMS), Jan 1, 1983

Research paper thumbnail of AMPL: A mathematical programming language

Computer science technical report, …, Jan 1, 1987

Research paper thumbnail of Solving symmetric indefinite systems in an interior-point method for linear programming

Mathematical Programming, Jan 1, 1993

Research paper thumbnail of Solving staircase linear programs by the simplex method, 1: Inversion

Mathematical Programming, Jan 1, 1982

Research paper thumbnail of A simplex algorithm for piecewise-linear programming I: Derivation and proof

Mathematical Programming, Jan 1, 1985

Research paper thumbnail of Optimization as an Internet resource

Research paper thumbnail of A survey of mathematical programming applications in integrated steel plants

Manufacturing & Service Operations …, Jan 1, 2001

Research paper thumbnail of A simplex algorithm for piecewise-linear programming II: Finiteness, feasibility and degeneracy

Mathematical Programming, Jan 1, 1988

Research paper thumbnail of Expressing complementarity problems in an algebraic modeling language and communicating them to solvers

SIAM Journal on Optimization, Jan 1, 1999

Research paper thumbnail of Constraints and AI planning

Intelligent Systems, …, Jan 1, 2005

Research paper thumbnail of A simplex algorithm for piecewise-linear programming III: Computational analysis and applications

Mathematical Programming, Jan 1, 1992

The first two parts of this paper have developed a simplex algorithm for minimizing convex separa... more The first two parts of this paper have developed a simplex algorithm for minimizing convex separable piecewise-linear functions subject to linear constraints. This concluding part argues that a direct piecewiselinear simplex implementation has inherent advantages over an indirect approach that relies on transformation to a linear program. The advantages are shown to be implicit in relationships between the linear and piecewise-linear algorithms, and to be independent of many details of implementation. Two sets of computational results serve to illustarate these arguments; the piecewise-linear simplex algorithm is observed to run 2–6 times faster than a comparable linear algorithm, not including any additional expense that might be incurred in setting up the equivalent linear program. Further support for the practical value of a good piecewise-linear programming algorithm is provided by a survey of many varied applications.

Research paper thumbnail of Database structures for mathematical programming models

Decision Support Systems, Jan 1, 1997

Research paper thumbnail of A study of the augmented system and column-splitting approaches for solving two-stage stochastic linear programs by interior-point methods

INFORMS Journal on …, Jan 1, 1995

Linear programs that arise in two-stage stochastic programming offer a particularly difficult tes... more Linear programs that arise in two-stage stochastic programming offer a particularly difficult test of the robustness of interior-point methods. These LPs are typically very large, yet incorporate "dense columns"—corresponding to the first-stage variables—that rule out the standard ...

Research paper thumbnail of Optimization on the NEOS Server

Research paper thumbnail of Expressing special structures in an algebraic modeling language for mathematical programming

ORSA Journal on Computing, Jan 1, 1995

Research paper thumbnail of Finding embedded network rows in linear programs I. Extraction heuristics

Management Science, Jan 1, 1988

Research paper thumbnail of Staircase matrices and systems

SIAM review, Jan 1, 1984

Abstract. Difference quations, spline approximations and multiperiod linear programs all give ris... more Abstract. Difference quations, spline approximations and multiperiod linear programs all give rise to linear equation systems that have a characteristic staircase structure. A staircase system's variables can be partitioned, into a natural sequence of periods, in such a way that every ...

Research paper thumbnail of Interdependence of Methods and Representations in Design of Software for Combinatorial Optimization

be entirely general (C, Fortran), or specialized for mathematical modeling (Matlab, Mathematica, ... more be entirely general (C, Fortran), or specialized for mathematical modeling (Matlab, Mathematica, Maple), or specialized for optimization as in the case of decades-old packages such as OMNI 29] and modern C++ libraries such as ILOG Solver 53]. The drawbacks of programming to describe optimization models are well known, however so-called \matrix generation" programs are hard to debug, to maintain, and to document 1 5 ]. Our focus in this paper is on higher-level representations that allow optimization models to be described non-procedurally, in terms familiar to human modelers. The continuing success of such representations | and of modeling systems based on them | testi es to their value in optimization. We survey in particular three representations popularly applied in combinatorial optimization, algebraic modeling languages, constraint logic programming languages, and netforms, or network diagrams. We rst describe the kinds of optimization methods and systems most commonly associated with these alternatives. Each p a i r o f r e p r e s e n tations is then considered, to show h o w each has been advantageous and how its advantages have begun to in uence (or ought to in uence) the design of the other. Our current research projects are described in conjunction with several of these comparisons.

Research paper thumbnail of Constraints and AI Planning

IEEE Intelligent Systems, 2005

Research paper thumbnail of A modeling language for mathematical programming

Management Science, Jan 1, 1990

Research paper thumbnail of Modeling languages versus matrix generators for linear programming

ACM Transactions on Mathematical Software (TOMS), Jan 1, 1983

Research paper thumbnail of AMPL: A mathematical programming language

Computer science technical report, …, Jan 1, 1987

Research paper thumbnail of Solving symmetric indefinite systems in an interior-point method for linear programming

Mathematical Programming, Jan 1, 1993

Research paper thumbnail of Solving staircase linear programs by the simplex method, 1: Inversion

Mathematical Programming, Jan 1, 1982

Research paper thumbnail of A simplex algorithm for piecewise-linear programming I: Derivation and proof

Mathematical Programming, Jan 1, 1985

Research paper thumbnail of Optimization as an Internet resource

Research paper thumbnail of A survey of mathematical programming applications in integrated steel plants

Manufacturing & Service Operations …, Jan 1, 2001

Research paper thumbnail of A simplex algorithm for piecewise-linear programming II: Finiteness, feasibility and degeneracy

Mathematical Programming, Jan 1, 1988

Research paper thumbnail of Expressing complementarity problems in an algebraic modeling language and communicating them to solvers

SIAM Journal on Optimization, Jan 1, 1999

Research paper thumbnail of Constraints and AI planning

Intelligent Systems, …, Jan 1, 2005

Research paper thumbnail of A simplex algorithm for piecewise-linear programming III: Computational analysis and applications

Mathematical Programming, Jan 1, 1992

The first two parts of this paper have developed a simplex algorithm for minimizing convex separa... more The first two parts of this paper have developed a simplex algorithm for minimizing convex separable piecewise-linear functions subject to linear constraints. This concluding part argues that a direct piecewiselinear simplex implementation has inherent advantages over an indirect approach that relies on transformation to a linear program. The advantages are shown to be implicit in relationships between the linear and piecewise-linear algorithms, and to be independent of many details of implementation. Two sets of computational results serve to illustarate these arguments; the piecewise-linear simplex algorithm is observed to run 2–6 times faster than a comparable linear algorithm, not including any additional expense that might be incurred in setting up the equivalent linear program. Further support for the practical value of a good piecewise-linear programming algorithm is provided by a survey of many varied applications.

Research paper thumbnail of Database structures for mathematical programming models

Decision Support Systems, Jan 1, 1997

Research paper thumbnail of A study of the augmented system and column-splitting approaches for solving two-stage stochastic linear programs by interior-point methods

INFORMS Journal on …, Jan 1, 1995

Linear programs that arise in two-stage stochastic programming offer a particularly difficult tes... more Linear programs that arise in two-stage stochastic programming offer a particularly difficult test of the robustness of interior-point methods. These LPs are typically very large, yet incorporate "dense columns"—corresponding to the first-stage variables—that rule out the standard ...

Research paper thumbnail of Optimization on the NEOS Server

Research paper thumbnail of Expressing special structures in an algebraic modeling language for mathematical programming

ORSA Journal on Computing, Jan 1, 1995

Research paper thumbnail of Finding embedded network rows in linear programs I. Extraction heuristics

Management Science, Jan 1, 1988

Research paper thumbnail of Staircase matrices and systems

SIAM review, Jan 1, 1984

Abstract. Difference quations, spline approximations and multiperiod linear programs all give ris... more Abstract. Difference quations, spline approximations and multiperiod linear programs all give rise to linear equation systems that have a characteristic staircase structure. A staircase system's variables can be partitioned, into a natural sequence of periods, in such a way that every ...