On a Production Allocation and Distribution Problem

In recent years many advances have been made in solution techniques for specially structured 0–1 integer programming problems. In contrast, very little progress has been made on solving general (mixed) integer problems. This, of course, is not true when viewed from the theoretical side: H. W. Lenstra (1983) made a major breakthrough, obtaining a polynomial-time algorithm when the number of integer variables is fixed. We discuss a practical implementation of a Lenstra-like algorithm, based on the generalized basis reduction method of L. Lovász and H. E. Scarf (1988). This method allows us to avoid the ellipsoidal approximations required in Lenstra's algorithm. We report on the solution of a number of small (but difficult) examples, with up to 100 integer variables. Our computer code uses the linear programming optimizer CPLEX as a subroutine to solve the linear programming problems that arise. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

In this paper, we propose a new hybrid social spider algorithm with simplex Nelder-Mead method in order to solve integer programming and minimax problems. We call the proposed algorithm a Simplex Social Spider optimization (SSSO) algorithm. In the the proposed SSSO algorithm, we combine the social spider algorithm with its powerful capability of performing exploration, exploitation, and the Nelder-Mead method in order to refine the best obtained solution from the standard social spider algorithm. In order to investigate the general performance of the proposed SSSO algorithm, we test it on 7 integer programming problems and 10 minimax problems and compare against 10 algorithms for solving integer programming problems and 9 algorithms for solving minimax problems. The experiments results show the efficiency of the proposed algorithm and its ability to solve integer and minimax optimization problems in reasonable time.

A Survey of Lagrangean Techniques for Discrete Optimization

Algorithms and computer-based tools for analyzing infeasible linear and nonlinear programs have been developed in recent years, but few such tools exist for infeasible mixed-integer or integer linear programs. One approach that has proven especially useful for infeasible linear programs is the isolation of an Irreducible Infeasible Set of constraints (IIS), a subset of the constraints defining the overall linear program that is itself infeasible, but for which any proper subset is feasible. Isolating an IIS from the larger model speeds the diagnosis and repair of the model by focussing the analytic effort. This paper describes and tests algorithms for finding small infeasible sets in infeasible mixed-integer and integer linear programs; where possible these small sets are IISs.

An approach for solving discrete optimization problems by decomposition is illustrated by a linear integer program and two integer problems. Numerical results are given for the inventory problems.

Ranking in quadratic integer programming problems

A brief review will be given of the computationel state of linear and integer programming. Difficulties in vectorization, techniques used, and performance improvements for linear programming will be presented. Integer programming is an even more computationally intensive problem area. Methods for reducing running times will be discussed, both in scalar and vector mode. The concept of improved linear programming formulations will be illustrated, and some cases discussed. Computational experience with parallel branching will be given. Other work in progress at the Watson Research Center in Yorktown will be reviewed. One area of work is the new, non-simplex approach based on Karmarkar’s work. Another is modeling tools and database connections.

Professional: comments on

We discuss an integer linear programming modeling system based on relational algebra. In this system, all modeling related activities, such as model formulation, model instantiation, and model and instance management, are done using simple operations such as selection, projection, and predicated join.

Previous article Next article Complementarity Theorems for Linear ProgrammingA. C. WilliamsA. C. Williamshttps://doi.org/10.1137/1012015PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Frank Eugene Clark, Mathematical Notes: Remark on the Constraint Sets in Linear Programming, Amer. Math. Monthly, 68 (1961), 351–352 MR1531192 0109.38204 CrossrefISIGoogle Scholar[2] A. J. Goldman, Resolution and separation theorems for polyhedral convex setsLinear inequalities and related systems, Annals of Mathematics Studies, no. 38, Princeton University Press, Princeton, N. J., 1956, 41–51, in [4] MR0089113 0072.37505 Google Scholar[3] A. J. Goldman and , A. W. Tucker, Theory of linear programmingLinear inequalities and related systems, Annals of Mathematics Studies, no. 38, Princeton University Press, Princeton, N.J., 1956, 53–97, in [4] MR0101826 0072.37601 Google Scholar[4] H. W. Kuhn and , A. W. Tucker, Linear Inequalities and Related Systems, Princeton University Press, Princeton, 1956 0072.37502 Google Scholar[5] A. C. Williams, Boundedness relations for linear constraint sets, Linear Algebra and Appl., 3 (1970), 129–141 10.1016/0024-3795(70)90009-1 MR0266612 0201.22003 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Sparse solutions to an underdetermined system of linear equations via penalized Huber loss6 November 2020 | Optimization and Engineering, Vol. 22, No. 3 Cross Ref Dynamic Non-diagonal Regularization in Interior Point Methods for Linear and Convex Quadratic Programming26 February 2019 | Journal of Optimization Theory and Applications, Vol. 181, No. 3 Cross Ref Necessary and Sufficient Conditions for Noiseless Sparse Recovery via Convex Quadratic SplinesMustafa Ç Pinar12 February 2019 | SIAM Journal on Matrix Analysis and Applications, Vol. 40, No. 1AbstractPDF (416 KB)A labeling algorithm for the sensitivity ranges of the assignment problemApplied Mathematical Modelling, Vol. 35, No. 10 Cross Ref Bibliography15 August 2011 Cross Ref Sensitivity Analysis and Dynamic Programming15 February 2011 Cross Ref Bibliography Cross Ref Condition measures and properties of the central trajectory of a linear programMathematical Programming, Vol. 83, No. 1-3 Cross Ref New characterizations of ℓ1 solutions to overdetermined systems of linear equationsOperations Research Letters, Vol. 16, No. 3 Cross Ref Limiting behavior of weighted central paths in linear programmingMathematical Programming, Vol. 65, No. 1-3 Cross Ref Stability of linearly constrained convex quadratic programsJournal of Optimization Theory and Applications, Vol. 64, No. 1 Cross Ref Marginal values in mixed integer linear programmingMathematical Programming, Vol. 44, No. 1-3 Cross Ref A theory of linear inequality systemsLinear Algebra and its Applications, Vol. 106 Cross Ref Boundedness relations in linear semi-infinite programmingAdvances in Applied Mathematics, Vol. 8, No. 1 Cross Ref A Variable-Complexity Norm Maximization ProblemO. L. Mangasarian and T. -H. Shiau17 July 2006 | SIAM Journal on Algebraic Discrete Methods, Vol. 7, No. 3AbstractPDF (698 KB)Simple computable bounds for solutions of linear complementarity problems and linear programs26 February 2009 Cross Ref On polyhedral extension of some LP theoremsMathematical Programming, Vol. 30, No. 1 Cross Ref Polyhedral extensions of some theorems of linear programmingMathematical Programming, Vol. 24, No. 1 Cross Ref Optimal simplex tableau characterization of unique and bounded solutions of linear programsJournal of Optimization Theory and Applications, Vol. 35, No. 1 Cross Ref Projection and Restriction Methods in Geometric Programming and Related Problems Cross Ref Representation of Convex Sets Cross Ref Projection and restriction methods in geometric programming and related problemsJournal of Optimization Theory and Applications, Vol. 26, No. 1 Cross Ref The complementary unboundedness of dual feasible solution sets in convex programmingMathematical Programming, Vol. 12, No. 1 Cross Ref Theorems on the dimension of convex setsLinear Algebra and its Applications, Vol. 12, No. 1 Cross Ref On the primal and dual constraint sets in geometric programmingJournal of Mathematical Analysis and Applications, Vol. 32, No. 3 Cross Ref Volume 12, Issue 1| 1970SIAM Review History Submitted:23 December 1968Published online:18 July 2006 InformationCopyright © 1970 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1012015Article page range:pp. 135-137ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

With the fast development of regenerative braking technologies in modern transportation systems, it has become popular to take into account the regenerated electric energy of electric vehicles for energy-saving purposes. In railway transportation, it was found that given the monotonicity of the vehicle speed during an acceleration or braking process, a partial speed optimisation model can be set up and solved by Mixed Integer Linear Programming. Taking into account the similarity between road traffic and rail transportation, this paper aims to build up a linear programming model to optimise the speed trajectory of an electric vehicle (EV) during eco-approach and departure (EAD) to achieve a minimum energy cost. Three cases have been studied. First, we consider an optimisation model when the preceding vehicle is at a full-stop status, for example, when it is at a road crossing. We set up a case scenario with a constant running distance but different running time when the following EV initiates the car-following process. We will further investigate if the following EV has to use up all available running time before it fully stops behind the preceding vehicle. Second, an optimisation model is proposed by predicting the movement of the preceding vehicle. In this way, we are considering an optimisation problem with varying distance and time for the target car. Third, we try to consider a case where the following EV tries to accelerate to the same speed of the preceding vehicle under the time and distance constraints. The motivation of this paper lies on the successful applications of linear programming for partial train speed trajectory optimisation, the capability of regenerative braking of plug-in all electric vehicles (PA-EV) and speed trajectory optimisation in the application EAD. The proposed model takes advantage of its robustness, computational efficiency and readiness of potential on-line energy-saving applications in intelligent and connected vehicle systems.

Introducción: Las decisiones operativas de un minorista implican que a través de su surtido se pueda satisfacer la demanda logrando la mayor utilidad posible haciendo el mejor uso posible de su espacio de exhibición. Este artículo presenta un modelo de programación lineal entera mixta cuyo propósito es maximizar la rentabilidad de un minorista considerando los efectos de sustitución dinámica entre artículos, impactos de la elasticidad del espacio sobre la demanda, niveles de servicio esperados de la exhibición y las decisiones de los compradores. Objetivo: Proponer un modelo que soporte la toma de decisiones de asignación de espacio, planeación del surtido y localización en góndola de un minorista. Metodología: Primero se analizan diferentes factores que afectan las decisiones objeto de estudio usando fuentes primarias y secundarias. Luego se desarrolla el modelo matemático y se evalúa su desempeño utilizando datos reales de una categoría de productos en un punto de venta de una cadena de supermercados. Resultados: En la instancia de prueba se obtiene la solución óptima en un tiempo computacional razonable y esta solución permite establecer la forma en la cual gestionar de mejor manera el surtido en el punto de venta, superando los resultados del método actual. Conclusiones: El modelo propuesto permite aumentar la utilidad de la cadena a través del surtido adecuado, con los espacios de exhibición justos y con la localización exacta de los artículos seleccionados.

Structured prediction is used in areas such as computer vision and natural language processing to predict structured outputs such as segmentations or parse trees. In these settings, prediction is performed by MAP inference or, equivalently, by solving an integer linear program. Because of the complex scoring functions required to obtain accurate predictions, both learning and inference typically require the use of approximate solvers. We propose a theoretical explanation to the striking observation that approximations based on linear programming (LP) relaxations are often tight on real-world instances. In particular, we show that learning with LP relaxed inference encourages integrality of training instances, and that tightness generalizes from train to test data.

Exact solution approaches for integer linear generalized maximum multiplicative programs through the lens of multi-objective optimization

Elementary (flux) modes (EMs) have served as a valuable tool for investigating structural and functional properties of metabolic networks. Identification of the full set of EMs in genome-scale networks remains challenging due to combinatorial explosion of EMs in complex networks. It is often, however, that only a small subset of relevant EMs needs to be known, for which optimization-based sequential computation is a useful alternative. Most of the currently available methods along this line are based on the iterative use of mixed integer linear programming (MILP), the effectiveness of which significantly deteriorates as the number of iterations builds up. To alleviate the computational burden associated with the MILP implementation, we here present a novel optimization algorithm termed alternate integer linear programming (AILP). Our algorithm was designed to iteratively solve a pair of integer programming (IP) and linear programming (LP) to compute EMs in a sequential manner. In each step, the IP identifies a minimal subset of reactions, the deletion of which disables all previously identified EMs. Thus, a subsequent LP solution subject to this reaction deletion constraint becomes a distinct EM. In cases where no feasible LP solution is available, IP-derived reaction deletion sets represent minimal cut sets (MCSs). Despite the additional computation of MCSs, AILP achieved significant time reduction in computing EMs by orders of magnitude. The proposed AILP algorithm not only offers a computational advantage in the EM analysis of genome-scale networks, but also improves the understanding of the linkage between EMs and MCSs. The software is implemented in Matlab, and is provided as supplementary information . hyunseob.song@pnnl.gov. Supplementary data are available at Bioinformatics online.