Branch and Cut

Mitchell, John E.

doi:10.1002/9780470400531.eorms0117

“…The main premise behind branch-and-cut [36] is that if the convex hull 1 of an MILP problem is obtained, the solution process is reduced to solving an LP problem. Owing to the linearity of the problem, the surface of the convex hull is polyhedral [41], and vertices of the convex hull are feasible solutions to the original MILP problem.…”

Section: Milp Method: Branch-and-cutmentioning

confidence: 99%

Distributed and Asynchronous Coordination of a Mixed-Integer Linear System via Surrogate Lagrangian Relaxation

Bragin¹,

Yan²,

Luh³

2021

Preprint

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With the emergence of Internet of Things that allows communications and local computations, and with the vision of Industry 4.0, a foreseeable transition is from centralized system planning and operation toward decentralization with interacting components and subsystems, e.g., self-optimizing factories. In this paper, a new “price-based” decomposition and coordination methodology is developed to efficiently coordinate subsystems such as machines and parts, which are described by Mixed-Integer Linear Programming (MILP) formulations, in a distributed and asynchronous way. To ensure low communication requirements, exchanges between the “coordinator” and subsystems are limited to “prices” (Lagrangian multipliers) broadcast by the coordinator, and to subsystem solutions sent to the coordinator. Asynchronous coordination, however, may lead to convergence difficulties since the order in which subsystem solutions arrive at the coordinator is not predefined as a result of uncertainties in communication and solving times. Under realistic assumptions of finite communication and solve times, convergence of our method is proved by innovatively extending Lyapunov Stability Theory. Numerical testing of generalized assignment problems through simulation demonstrates that the method converges fast and provides near-optimal results, paving the way for self-optimizing factories in the future.

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“…The main premise behind branch-and-cut [36] is that if the convex hull 1 of an MILP is obtained, the problem reduces to solving a linear programming problem. Owing to linearity of the problem, the surface of the convex hull is polyhedral [41], and vertices of the convex hull are feasible solutions to the original MILP problem.…”

Section: Milp Method: Branch-and-cutmentioning

confidence: 99%

Asynchronous Coordination of Distributed Mixed-Integer Linear Subsystems via Surrogate Lagrangian Relaxation

Bragin¹,

Luh²,

Yan³

2020

Preprint

2

0

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With the emergence of Internet of Things that allows communications and local computations, and with the vision of Industry 4.0, a foreseeable transition is from centralized system planning and operation toward decentralization with interacting components and subsystems, e.g., self-optimizing factories. In this paper, a new “price-based” decomposition and coordination methodology is developed to efficiently coordinate subsystems such as machines and parts, which are described by Mixed-Integer Linear Programming (MILP) formulations, in a distributed and asynchronous way. To ensure low communication requirements, exchanges between the “coordinator” and subsystems are limited to “prices” (Lagrangian multipliers) broadcast by the coordinator, and to subsystem solutions sent to the coordinator. Asynchronous coordination, however, may lead to convergence difficulties since the order in which subsystem solutions arrive at the coordinator is not predefined as a result of uncertainties in communication and solving times. Under realistic assumptions of finite communication and solve times, convergence of our method is proved by innovatively extending Lyapunov Stability Theory. Numerical testing of generalized assignment problems through simulation demonstrates that the method converges fast and provides near-optimal results, paving the way for self-optimizing factories in the future.

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“…However, there are many cases in which all known MIP formulations (strong or weak) are large. If these formulations are such that the number of variables is small and the number of constraints is large, but can be separated fast, it is sometimes possible to use them in a branch-andcut procedure [131,39,128]. Similarly, when only the number of variables is large, the formulations can be used in column generation or branch-and-price procedures [15].…”

Section: Large Formulationsmentioning

confidence: 99%

Mixed Integer Linear Programming Formulation Techniques

Vielma¹

2015

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1. Introduction. Throughout more than 50 years of existence, mixed integer linear programming (MIP) theory and practice have been significantly developed, and it is now an indispensable tool in business and engineering [68,94,104]. Two reasons for the success of MIP are linear programming (LP) based solvers and the modeling flexibility of MIP. We now have several extremely effective state-of-the-art solvers [82,69, 52,171] that incorporate many advanced techniques [1,2,25,23,92,112,24] and, since its early stages, MIP has been used to model a wide range of applications [44,45].While in many cases constructing valid MIP formulations is relatively straightforward, some care should be taken in this construction as certain formulation attributes can significantly reduce the effectiveness of LP-based solvers. Fortunately, constructing formulations that behave well with state-of-the-art solvers can usually be achieved by following simple guidelines described in standard textbooks. However, more advanced techniques can often perform significantly better than textbook formulations and are sometimes a necessity. The main objective of this survey is to summarize the state of the art of such formulation techniques for a wide range of problems. To keep the length of this survey under control, we concentrate on formulations for sets of a mixed integer nature that require both integer constrained and continuous variables. We hence purposefully place less emphasis on some related areas such as combinatorial optimization, quadratic and polynomial 0/1 optimization, and polyhedral approximations of convex sets. These topics are certainly areas of important and active research, so we cover them succinctly in section 12.

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Branch and Cut

Cited by 21 publications

References 67 publications

Distributed and Asynchronous Coordination of a Mixed-Integer Linear System via Surrogate Lagrangian Relaxation

Distributed and Asynchronous Coordination of a Mixed-Integer Linear System via Surrogate Lagrangian Relaxation

Asynchronous Coordination of Distributed Mixed-Integer Linear Subsystems via Surrogate Lagrangian Relaxation

Mixed Integer Linear Programming Formulation Techniques

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