18. Mixed-Integer Programming: A Progress Report

Bixby, Robert E.; Fenelon, Mary; Gu, Zonghao; Rothberg, Edward; Wunderling, Roland

doi:10.1137/1.9780898718805.ch18

Cited by 88 publications

(49 citation statements)

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“…Split cuts [12], Gomory Mixed Integer (GMI) cuts [17], and Mixed Integer Rounding (MIR) cuts [23,24] are some of the most effective valid inequalities for Mixed Integer Linear Programming (MILP) [8]. While they are known to be equivalent [15,24], each of them provide different advantages and insights.…”

Section: Introductionmentioning

confidence: 99%

Split cuts and extended formulations for Mixed Integer Conic Quadratic Programming

Modaresi

Kılınç

Vielma

2015

Operations Research Letters

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We study split cuts and extended formulations for Mixed Integer Conic Quadratic Programming (MICQP) and their relation to Conic Mixed Integer Rounding (CMIR) cuts. We show that CMIR is a linear split cut for the polyhedral portion of an extended formulation of a quadratic set and it can be weaker than the nonlinear split cut of the same quadratic set. However, we also show that families of CMIRs can be significantly stronger than the associated family of nonlinear split cuts.

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Section: Introductionmentioning

confidence: 99%

Split cuts and extended formulations for Mixed Integer Conic Quadratic Programming

Modaresi

Kılınç

Vielma

2015

Operations Research Letters

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“…As linear programming is at the heart of branch-and-bound methods for integer programming, we observe that Bixby et al [14] report a 2360 fold speed-up of the CPLEX linear programming code from 1988 to 2002 and that, in the same period of time, an additional 800 fold speed-up is obtained due to advances in hardware.…”

Section: Success Storiesmentioning

confidence: 66%

“…Our first example ( [14]) is a weekly production planning model using daily buckets. The objective is to minimize end-of-day inventory.…”

Section: Success Storiesmentioning

confidence: 99%

Integer-Programming Software Systems

2005

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Abstract. Recent developments in integer-programming software systems have tremendously improved our ability to solve large-scale instances. We review the major algorithmic components of state-of-the-art solvers and discuss the options available to users to adjust the behavior of these solvers when default settings do not achieve the desired performance level. Furthermore, we highlight advances towards integrated modeling and solution environments. We conclude with a discussion of model characteristics and substructures that pose challenges for integer-programming software systems and a perspective on features we may expect to see in these systems in the near future.

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“…A common approach to try to address bigger problems is to do an implicit enumeration via a branch-and-bound algorithm [24,25]. Branch-and-bound is a standard algorithm for integer programming (see, for example, [26] and references therein). A brief sketch of it in the context of this paper is given next, with an emphasis on the particularities that arise in the context of multiphase MIOCP.…”

Section: Minlp Resolutionmentioning

confidence: 99%

Multiphase Mixed-Integer Optimal Control Approach to Aircraft Trajectory Optimization

Bonami

Olivares

Soler

et al. 2013

Journal of Guidance, Control, and Dynamics

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In this paper, an approach to aircraft trajectory optimization is presented in which integer and continuous variables are considered. Integer variables model decision-making processes, and continuous variables describe the state of the aircraft, which evolves according to differential-algebraic equations. The problem is formulated as a multiphase mixed-integer optimal control problem. It is transcribed into a mixed-integer nonlinear programming problem by applying a fifth degree Gauss-Lobatto direct collocation method and is then solved using a nonlinear-programming-based branch-and-bound algorithm. The approach is applied to the following en route flight planning problem: Given an aircraft point mass model, a wind forecast, an airspace structure, and the relevant flying information regions with their associated overflying costs, find the control inputs that steer the aircraft from the initial fix to the final fix, following a route of waypoints while minimizing the fuel consumption and overflying costs during the flight. The decision-making process arises in determining the optimal sequence of waypoints. The optimal times at which the waypoints are to be overflown are also to be determined. Numerical results are presented and discussed, showing the effectiveness of the approach.

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18. Mixed-Integer Programming: A Progress Report

Cited by 88 publications

References 0 publications

Split cuts and extended formulations for Mixed Integer Conic Quadratic Programming

Split cuts and extended formulations for Mixed Integer Conic Quadratic Programming

Integer-Programming Software Systems

Multiphase Mixed-Integer Optimal Control Approach to Aircraft Trajectory Optimization

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