International Journal of Information Technology and Decision Making, Aug 11, 2021
Finding good solutions to clique partitioning problems remains a computational challenge. With ra... more Finding good solutions to clique partitioning problems remains a computational challenge. With rare exceptions, finding optimal solutions for all but small instances is not practically possible. However, choosing the most appropriate modeling structure can have a huge impact on what is practical to obtain from exact solvers within a reasonable amount of run time. Commercial solvers have improved tremendously in recent years and the combination of the right solver and the right model can significantly increase our ability to compute acceptable solutions to modest-sized problems with solvers like CPLEX, GUROBI and XPRESS. In this paper, we explore and compare the use of three commercial solvers on modest sized test problems for clique partitioning. For each problem instance, a conventional linear model from the literature and a relatively new quadratic model are compared. Extensive computational experience indicates that the quadratic model outperforms the classic linear model as problem size grows.
We propose a new algorithm for fixed‐charge network flow problems based on ghost image (GI) proce... more We propose a new algorithm for fixed‐charge network flow problems based on ghost image (GI) processes as proposed in Glover (1994) and adapted to fixed‐charge transportation problems in Glover et al. (2005). Our GI algorithm iteratively modifies an idealized representation of the problem embodied in a parametric GI, enabling all steps to be performed with a primal network flow algorithm operating on the parametric GI. Computational testing is carried out on well‐known problems from the literature plus a new set of large‐scale fixed‐charge transportation and transshipment network instances. We also provide comparisons against CPLEX 12.8 and demonstrate that the new GI algorithm with tabu search (TS) is effective on large problem instances, finding solutions with statistically equivalent objective values at least 700 times faster. The attractive outcomes produced by the current GI/TS implementation provide a significant advance in our ability to solve fixed‐cost network problems effic...
This article shows how simple systems of linear equations with {0,1} variables can be aggregated ... more This article shows how simple systems of linear equations with {0,1} variables can be aggregated into a single linear equation whose {0,1} solutions are identical to the solutions of the original system. Structures of the original systems are exploited to keep the aggregator's integer coefficients from becoming unnecessarily large. The results have potential application in integer programming and information theory, especially for problems that contain assignment-type constraints along with other constraints. Several unresolved questions of a number-theoretic nature are mentioned at the conclusion of the article.
This note develops efficient sensitivity procedures for posynomial geometric programs. These proc... more This note develops efficient sensitivity procedures for posynomial geometric programs. These procedures provide ranging information for the primal coefficients, means for dealing with problems with loose primal constraints, and an incremental procedure for improving the estimated solutions. These sensitivity procedures are independent of the method of solution of the geometric program.
This paper presents an implementation of surrogate constraint duality in mathematical programming... more This paper presents an implementation of surrogate constraint duality in mathematical programming. Motivated by the use of linear programming duality for surrogate constraints in integer linear programs, this implementation is based on geometric programming duality. As a result of this formulation we are able to present an algorithm for surrogate constraint duality and discuss several important properties of the algorithm.
This paper presents an implementation of some recent results of Bigelow and Shapiro [1]. These im... more This paper presents an implementation of some recent results of Bigelow and Shapiro [1]. These implicit function theorems are shown to provide a convenient means of performing certain types of sensitivity analysis, in particular updating the lagrange multipliers, associated with particular classes of problems. As a result we extend the usual sensitivity analysis results to include improving estimates of the effect of changing the right-hand sides of constraints. Examples of chemical equilibrium and entropy maximization models are used to illustrate the results.
The paper presents an analysis of the constrained entropy maximization model from the point of vi... more The paper presents an analysis of the constrained entropy maximization model from the point of view of geometric programming. While the original entropy maximization model consists of maximizing the entropy of a system subject only to constraints that the solution be a probability measure, the models considered here contain an additional set of linear constraints. These constrained models have been the subject of a wide range of applications in transportation and geographical analysis. Using the duality theory of geometric programming, we develop the dual to the constrained model, which as in the case of the original model is unconstrained except for the positivity restrictions on the dual variables. In addition, this duality theory enables us to study the solvability of the model and the impact of changes in the model parameters on the solution. The sensitivity analysis provides approximations to the optimal solution to problems with perturbed data without requiring the re-solving ...
The Journal of the Operational Research Society, 1994
ABSTRACT In this note, single machine scheduling and minimizing absolute flow time deviation (TAF... more ABSTRACT In this note, single machine scheduling and minimizing absolute flow time deviation (TAFD) are considered. The relationship between this problem and the mean absolute deviation of job completion times about a common due date (MAD) is discussed. Based on the MAD problem optimal solutions of the TAFD problem are given. Furthermore, the generalization of the problem to the multiple machine case is discussed and an efficient algorithm for generating many optimal solutions of the problem, in the multi-machine case, is given.
Summary A method of obtaining discrete and/or integer valued solutions to non-linear design probl... more Summary A method of obtaining discrete and/or integer valued solutions to non-linear design problems is presented. The general framework is that of geometric programming which is combined with the Branch and Bound Method. Recently developed computational procedures are described and are used to demonstrate the feasibility of the above method.
International Journal of Mathematical Modelling and Numerical Optimisation, 2009
... He is currently a Professor of Operations Management at the School of Business Administration... more ... He is currently a Professor of Operations Management at the School of Business Administration, the University of Mississippi. ... Cornuéjols and Dawande (1999) (C&D) pointed out that even small instances of these problems (30 variables and a few constraints) are extremely ...
International Journal of Information Technology & Decision Making, 2013
In this paper, the cardinality constrained quadratic model for binary quadratic programming is us... more In this paper, the cardinality constrained quadratic model for binary quadratic programming is used to model and solve the graph bisection problem as well as its generalization in the form of the task allocation problem with two processors (2-TAP). Balanced graph bisection is an NP-complete problem which partitions a set of nodes in the graph G = (N, E) into two sets with equal cardinality such that a minimal sum of edge weights exists between the nodes in the two separate sets. 2-TAP is graph bisection with the addition of node preference costs in the objective function. We transform the general linear k-TAP model to the cardinality constrained quadratic binary model so that it may be efficiently solved using tabu search with strategic oscillation. On a set of benchmark graph bisections, we improve the best known solution for several problems. Comparison results with the state-of-the-art graph partitioning program METIS, as well as Cplex and Gurobi are presented on a set of randoml...
International Journal of Information Technology and Decision Making, Aug 11, 2021
Finding good solutions to clique partitioning problems remains a computational challenge. With ra... more Finding good solutions to clique partitioning problems remains a computational challenge. With rare exceptions, finding optimal solutions for all but small instances is not practically possible. However, choosing the most appropriate modeling structure can have a huge impact on what is practical to obtain from exact solvers within a reasonable amount of run time. Commercial solvers have improved tremendously in recent years and the combination of the right solver and the right model can significantly increase our ability to compute acceptable solutions to modest-sized problems with solvers like CPLEX, GUROBI and XPRESS. In this paper, we explore and compare the use of three commercial solvers on modest sized test problems for clique partitioning. For each problem instance, a conventional linear model from the literature and a relatively new quadratic model are compared. Extensive computational experience indicates that the quadratic model outperforms the classic linear model as problem size grows.
We propose a new algorithm for fixed‐charge network flow problems based on ghost image (GI) proce... more We propose a new algorithm for fixed‐charge network flow problems based on ghost image (GI) processes as proposed in Glover (1994) and adapted to fixed‐charge transportation problems in Glover et al. (2005). Our GI algorithm iteratively modifies an idealized representation of the problem embodied in a parametric GI, enabling all steps to be performed with a primal network flow algorithm operating on the parametric GI. Computational testing is carried out on well‐known problems from the literature plus a new set of large‐scale fixed‐charge transportation and transshipment network instances. We also provide comparisons against CPLEX 12.8 and demonstrate that the new GI algorithm with tabu search (TS) is effective on large problem instances, finding solutions with statistically equivalent objective values at least 700 times faster. The attractive outcomes produced by the current GI/TS implementation provide a significant advance in our ability to solve fixed‐cost network problems effic...
This article shows how simple systems of linear equations with {0,1} variables can be aggregated ... more This article shows how simple systems of linear equations with {0,1} variables can be aggregated into a single linear equation whose {0,1} solutions are identical to the solutions of the original system. Structures of the original systems are exploited to keep the aggregator's integer coefficients from becoming unnecessarily large. The results have potential application in integer programming and information theory, especially for problems that contain assignment-type constraints along with other constraints. Several unresolved questions of a number-theoretic nature are mentioned at the conclusion of the article.
This note develops efficient sensitivity procedures for posynomial geometric programs. These proc... more This note develops efficient sensitivity procedures for posynomial geometric programs. These procedures provide ranging information for the primal coefficients, means for dealing with problems with loose primal constraints, and an incremental procedure for improving the estimated solutions. These sensitivity procedures are independent of the method of solution of the geometric program.
This paper presents an implementation of surrogate constraint duality in mathematical programming... more This paper presents an implementation of surrogate constraint duality in mathematical programming. Motivated by the use of linear programming duality for surrogate constraints in integer linear programs, this implementation is based on geometric programming duality. As a result of this formulation we are able to present an algorithm for surrogate constraint duality and discuss several important properties of the algorithm.
This paper presents an implementation of some recent results of Bigelow and Shapiro [1]. These im... more This paper presents an implementation of some recent results of Bigelow and Shapiro [1]. These implicit function theorems are shown to provide a convenient means of performing certain types of sensitivity analysis, in particular updating the lagrange multipliers, associated with particular classes of problems. As a result we extend the usual sensitivity analysis results to include improving estimates of the effect of changing the right-hand sides of constraints. Examples of chemical equilibrium and entropy maximization models are used to illustrate the results.
The paper presents an analysis of the constrained entropy maximization model from the point of vi... more The paper presents an analysis of the constrained entropy maximization model from the point of view of geometric programming. While the original entropy maximization model consists of maximizing the entropy of a system subject only to constraints that the solution be a probability measure, the models considered here contain an additional set of linear constraints. These constrained models have been the subject of a wide range of applications in transportation and geographical analysis. Using the duality theory of geometric programming, we develop the dual to the constrained model, which as in the case of the original model is unconstrained except for the positivity restrictions on the dual variables. In addition, this duality theory enables us to study the solvability of the model and the impact of changes in the model parameters on the solution. The sensitivity analysis provides approximations to the optimal solution to problems with perturbed data without requiring the re-solving ...
The Journal of the Operational Research Society, 1994
ABSTRACT In this note, single machine scheduling and minimizing absolute flow time deviation (TAF... more ABSTRACT In this note, single machine scheduling and minimizing absolute flow time deviation (TAFD) are considered. The relationship between this problem and the mean absolute deviation of job completion times about a common due date (MAD) is discussed. Based on the MAD problem optimal solutions of the TAFD problem are given. Furthermore, the generalization of the problem to the multiple machine case is discussed and an efficient algorithm for generating many optimal solutions of the problem, in the multi-machine case, is given.
Summary A method of obtaining discrete and/or integer valued solutions to non-linear design probl... more Summary A method of obtaining discrete and/or integer valued solutions to non-linear design problems is presented. The general framework is that of geometric programming which is combined with the Branch and Bound Method. Recently developed computational procedures are described and are used to demonstrate the feasibility of the above method.
International Journal of Mathematical Modelling and Numerical Optimisation, 2009
... He is currently a Professor of Operations Management at the School of Business Administration... more ... He is currently a Professor of Operations Management at the School of Business Administration, the University of Mississippi. ... Cornuéjols and Dawande (1999) (C&D) pointed out that even small instances of these problems (30 variables and a few constraints) are extremely ...
International Journal of Information Technology & Decision Making, 2013
In this paper, the cardinality constrained quadratic model for binary quadratic programming is us... more In this paper, the cardinality constrained quadratic model for binary quadratic programming is used to model and solve the graph bisection problem as well as its generalization in the form of the task allocation problem with two processors (2-TAP). Balanced graph bisection is an NP-complete problem which partitions a set of nodes in the graph G = (N, E) into two sets with equal cardinality such that a minimal sum of edge weights exists between the nodes in the two separate sets. 2-TAP is graph bisection with the addition of node preference costs in the objective function. We transform the general linear k-TAP model to the cardinality constrained quadratic binary model so that it may be efficiently solved using tabu search with strategic oscillation. On a set of benchmark graph bisections, we improve the best known solution for several problems. Comparison results with the state-of-the-art graph partitioning program METIS, as well as Cplex and Gurobi are presented on a set of randoml...
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