12 research outputs found
Searching for optimal integer solutions to set partitioning problems using column generation
We describe a new approach to produce integer feasible columns to a set partitioning problem directly in
solving the linear programming (LP) relaxation using column generation. Traditionally, column generation
is aimed to solve the LP relaxation as quick as possible without any concern of the integer properties of
the columns formed. In our approach we aim to generate the columns forming the optimal integer solution
while simultaneously solving the LP relaxation. By this we can remove column generation in the branch
and bound search. The basis is a subgradient technique applied to a Lagrangian dual formulation of the set
partitioning problem extended with an additional surrogate constraint. This extra constraint is not relaxed
and is used to better control the subgradient evaluations. The column generation is then directed, via the
multipliers, to construct columns that form feasible integer solutions. Computational experiments show that
we can generate the optimal integer columns in a large set of well known test problems as compared to both
standard and stabilized column generation and simultaneously keep the number of columns smaller than
standard column generation
Modelo para la Localización de un Plataforma de Cross Docking en el Contexto de Logística Focalizada
The current context of global defence scenarios has enhanced development of new concepts like focused logistics concept. Focused logistics is a concept created to support logistics operations in uncertainty environments. In this framework focused logistics needs tools to enable an agile, reliable and accurate decision processes.
In order of these requirements, this paper shows a mathematical model as a support tool to find a good location of a cross-docking facility and allocate workloads of different kinds of vehicles from suppliers to demand centres being especially innovative to the strategy used for the solution of the present model.El actual contexto de los escenarios globales de defensa ha impulsado el desarrollo de conceptos cómo el de logística focalizada, creada para brindar soporte logístico en ambientes de alta volatilidad e incertidumbre Dentro de ese marco la logística focalizada requiere de herramientas que permitan agilizar los procesos de toma de decisiones. Atendiendo dicha necesidad se presenta un modelo que pretende soportar la decisión de localizaciónde una plataforma de cross-docking y asignación de medios de transporte que permita garantizar unadistribución ágil desde orígenes de suministro hasta los orígenes de demanda siendo especialmente novedosala estratégica de solución del mismo
Localización de una plataforma de cross docking en el contexto de logística focalizada utilizando distancias euclidianas
The current context of global defense scenarios has enhanced development of new concepts like focused logistics concept. Focused logistics is a concept created to supporrt logistics operations in uncertainty environments. In this framework focused logistics needs tools to enable an agile, reliable and accurate decision process. In order of these requirements this paper shows mathematical model to find a goog location of a cross-docking facility and allocate worlaoads of different kinds of vehicles from suppliers to demand centres using euclidean distances. This model is presented as a support tool in tactical operation being especially innovarive to the strategy used for the solution of the model.El actual contexto de los escenarios globales de defensa ha impulsado el desarrollo de conceptos cómo el de logística focalizada, creado para brindar soporte logístico en ambientes de alta volatilidad e incertidumbre. Dentro de ese marco la logística focalizada requiere de herramientas que permitan agilizar los procesos de toma de decisiones en donde la modelación matemática es útil como herramienta de soporte. En orden a lo anterior, se presenta un modelo que pretende soportar la decisión de localización de una plataforma de cross-docking y asignación de medios de transporte utilizando distancias euclidianas de tal manera que se permita garantizar una distribución ágil desde orígenes de suministro hasta los orígenes de demanda siendo especialmente novedosa la estratégica de solución del mismo
Using the primal-dual interior point algorithm within the branch-price-and-cut method
AbstractBranch-price-and-cut has proven to be a powerful method for solving integer programming problems. It combines decomposition techniques with the generation of both columns and valid inequalities and relies on strong bounds to guide the search in the branch-and-bound tree. In this paper, we present how to improve the performance of a branch-price-and-cut method by using the primal-dual interior point algorithm. We discuss in detail how to deal with the challenges of using the interior point algorithm with the core components of the branch-price-and-cut method. The effort to overcome the difficulties pays off in a number of advantageous features offered by the new approach. We present the computational results of solving well-known instances of the vehicle routing problem with time windows, a challenging integer programming problem. The results indicate that the proposed approach delivers the best overall performance when compared with a similar branch-price-and-cut method which is based on the simplex algorithm
Design and implementation of a modular interior-point solver for linear optimization
This paper introduces the algorithmic design and implementation of Tulip, an
open-source interior-point solver for linear optimization. It implements a
regularized homogeneous interior-point algorithm with multiple centrality
corrections, and therefore handles unbounded and infeasible problems. The
solver is written in Julia, thus allowing for a flexible and efficient
implementation: Tulip's algorithmic framework is fully disentangled from linear
algebra implementations and from a model's arithmetic. In particular, this
allows to seamlessly integrate specialized routines for structured problems.
Extensive computational results are reported. We find that Tulip is competitive
with open-source interior-point solvers on the H. Mittelmann's benchmark of
barrier linear programming solvers. Furthermore, we design specialized linear
algebra routines for structured master problems in the context of Dantzig-Wolfe
decomposition. These routines yield a tenfold speedup on large and dense
instances that arise in power systems operation and two-stage stochastic
programming, thereby outperforming state-of-the-art commercial interior point
method solvers. Finally, we illustrate Tulip's ability to use different levels
of arithmetic precision by solving problems in extended precision
Implementation of warm-start strategies in interior-point methods for linear programming in fixed dimension
We implement several warm-start strategies in interior-point methods for linear programming (LP). We study the situation in which both the original LP instance and the perturbed one have exactly the same dimensions. We consider different types of perturbations of data components of the original instance and different sizes of each type of perturbation. We modify the state-of-the-art interior-point solver PCx in our implementation. We evaluate the effectiveness of each warm-start strategy based on the number of iterations and the computation time in comparison with "cold start" on the NETLIB test suite. Our experiments reveal that each of the warm-start strategies leads to a reduction in the number of interior-point iterations especially for smaller perturbations and for perturbations of fewer data components in comparison with cold start. On the other hand, only one of the warm-start strategies exhibits better performance than cold start in terms of computation time. Based on the insight gained from the computational results, we discuss several potential improvements to enhance the performances of such warm-start strategies. © 2007 Springer Science+Business Media, LLC
A branch, price, and cut approach to solving the maximum weighted independent set problem
The maximum weight-independent set problem (MWISP) is one of the most
well-known and well-studied NP-hard problems in the field of combinatorial
optimization.
In the first part of the dissertation, I explore efficient branch-and-price (B&P)
approaches to solve MWISP exactly. B&P is a useful integer-programming tool for
solving NP-hard optimization problems. Specifically, I look at vertex- and edge-disjoint
decompositions of the underlying graph. MWISPâÂÂs on the resulting subgraphs are less
challenging, on average, to solve. I use the B&P framework to solve MWISP on the
original graph G using these specially constructed subproblems to generate columns. I
demonstrate that vertex-disjoint partitioning scheme gives an effective approach for
relatively sparse graphs. I also show that the edge-disjoint approach is less effective than
the vertex-disjoint scheme because the associated DWD reformulation of the latter
entails a slow rate of convergence.
In the second part of the dissertation, I address convergence properties associated
with Dantzig-Wolfe Decomposition (DWD). I discuss prevalent methods for improving the rate of convergence of DWD. I also implement specific methods in application to the
edge-disjoint B&P scheme and show that these methods improve the rate of
convergence.
In the third part of the dissertation, I focus on identifying new cut-generation
methods within the B&P framework. Such methods have not been explored in the
literature. I present two new methodologies for generating generic cutting planes within
the B&P framework. These techniques are not limited to MWISP and can be used in
general applications of B&P. The first methodology generates cuts by identifying faces
(facets) of subproblem polytopes and lifting associated inequalities; the second
methodology computes Lift-and-Project (L&P) cuts within B&P. I successfully
demonstrate the feasibility of both approaches and present preliminary computational
tests of each
A branch, price, and cut approach to solving the maximum weighted independent set problem
The maximum weight-independent set problem (MWISP) is one of the most
well-known and well-studied NP-hard problems in the field of combinatorial
optimization.
In the first part of the dissertation, I explore efficient branch-and-price (B&P)
approaches to solve MWISP exactly. B&P is a useful integer-programming tool for
solving NP-hard optimization problems. Specifically, I look at vertex- and edge-disjoint
decompositions of the underlying graph. MWISPâÂÂs on the resulting subgraphs are less
challenging, on average, to solve. I use the B&P framework to solve MWISP on the
original graph G using these specially constructed subproblems to generate columns. I
demonstrate that vertex-disjoint partitioning scheme gives an effective approach for
relatively sparse graphs. I also show that the edge-disjoint approach is less effective than
the vertex-disjoint scheme because the associated DWD reformulation of the latter
entails a slow rate of convergence.
In the second part of the dissertation, I address convergence properties associated
with Dantzig-Wolfe Decomposition (DWD). I discuss prevalent methods for improving the rate of convergence of DWD. I also implement specific methods in application to the
edge-disjoint B&P scheme and show that these methods improve the rate of
convergence.
In the third part of the dissertation, I focus on identifying new cut-generation
methods within the B&P framework. Such methods have not been explored in the
literature. I present two new methodologies for generating generic cutting planes within
the B&P framework. These techniques are not limited to MWISP and can be used in
general applications of B&P. The first methodology generates cuts by identifying faces
(facets) of subproblem polytopes and lifting associated inequalities; the second
methodology computes Lift-and-Project (L&P) cuts within B&P. I successfully
demonstrate the feasibility of both approaches and present preliminary computational
tests of each