Incremental and encoding formulations for Mixed Integer Programming

Yıldız, Sercan; Vielma, Juan Pablo

doi:10.1016/j.orl.2013.09.004

Cited by 19 publications

(10 citation statements)

References 28 publications

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“…This is illustrated in the following example from [168], which shows that variable branching on the w variables can be much more effective than variable branching on the y variables.…”

Section: Incremental Formulationsmentioning

confidence: 99%

“…We can show that a pure branch-and-bound algorithm requires branching in at least k/2 of the variables of (8.1) to solve it (see [168] for more details). However, a pure branch-and-bound algorithm can solve (8.2) branching in a single variable as follows.…”

Section: Incremental Formulationsmentioning

confidence: 99%

“…However, in some cases, an additional transformation of the continuous variables can result in an ad-hoc incremental formulation. An example of this is the following formulation introduced in [168] that generalizes a formulation for piecewise linear functions introduced in [165]. that satisfy a special ordering condition, formulation (8.3) reduces to a standard MIP formulation for piecewise linear functions that is denoted the incremental model in [155].…”

Section: Incremental Formulationsmentioning

confidence: 99%

“…that satisfy a special ordering condition, formulation (8.3) reduces to a standard MIP formulation for piecewise linear functions that is denoted the incremental model in [155]. For more details, examples of incremental formulations, and their advantages, we refer the reader to [168,20].…”

Section: Incremental Formulationsmentioning

confidence: 99%

“…The resulting formulation was introduced in [155] for piecewise linear functions and its extension to general polyhedra was introduced in [168].…”

Section: Incremental Formulationsmentioning

confidence: 99%

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Mixed Integer Linear Programming Formulation Techniques

Vielma¹

2015

SIAM Rev.

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268

143

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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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“…This is illustrated in the following example from [168], which shows that variable branching on the w variables can be much more effective than variable branching on the y variables.…”

Section: Incremental Formulationsmentioning

confidence: 99%

Section: Incremental Formulationsmentioning

confidence: 99%

Section: Incremental Formulationsmentioning

confidence: 99%

Section: Incremental Formulationsmentioning

confidence: 99%

“…The resulting formulation was introduced in [155] for piecewise linear functions and its extension to general polyhedra was introduced in [168].…”

Section: Incremental Formulationsmentioning

confidence: 99%

See 3 more Smart Citations

Mixed Integer Linear Programming Formulation Techniques

Vielma¹

2015

SIAM Rev.

Self Cite

268

143

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New multi-commodity flow formulations for the pooling problem

2016

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The pooling problem is a nonconvex nonlinear programming problem with numerous applications. The nonlinearities of the problem arise from bilinear constraints that capture the blending of raw materials. Bilinear constraints are well-studied and significant progress has been made in solving large instances of the pooling problem to global optimality. This is due in no small part to reformulations of the problem. Recently, Alfaki and Haugland proposed a multi-commodity flow formulation of the pooling problem based on input commodities. The authors proved that the new formulation has a stronger linear relaxation than previously known formulations. They also provided computational results which show that the new formulation outperforms previously known formulations when used in a global optimization solver. In this paper, we generalize their ideas and propose new multi-commodity flow formulations based on output, input and output and (input, output)-commodities. We prove the equivalence of formulations, and we study the partial order of formulations with respect to the strength of their LP relaxations. In an extensive computational study, we evaluate the performance of the new formulations. We study the trade-off between disaggregating commodities and therefore increasing the size of formulations versus strengthening the relaxed linear programs and improving the computational performance of the nonlinear programs. We provide computational results which show that output commodities often outperform input commodities, and that disaggregating commodities further only marginally strengthens the linear relaxations. In fact, smaller formulations often show a significantly better performance when used in a global optimization solver.

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An optimization approach for winner determination problem considering transportation cost discounts

Fang

Huang

2021

J Glob Optim

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Incremental and encoding formulations for Mixed Integer Programming

Cited by 19 publications

References 28 publications

Mixed Integer Linear Programming Formulation Techniques

Mixed Integer Linear Programming Formulation Techniques

New multi-commodity flow formulations for the pooling problem

An optimization approach for winner determination problem considering transportation cost discounts

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