Using Piecewise Linear Functions for Solving MINLPs

Geißler, Björn; Martín, Alexander; Morsi, Antonio; Schewe, Lars

doi:10.1007/978-1-4614-1927-3_10

Cited by 104 publications

(91 citation statements)

References 24 publications

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“…For example, [5,16] present computational results which show that formulations for piecewiselinear functions based on the binary encoding can significantly outperform those based on the unary and incremental encodings. In contrast, in a di↵erent set of problems, [32] reports that the incremental formulation performs better than the binary encoding formulation. Formulation (7) o↵ers a convenient way to preliminarily evaluate the performance of these three encodings without having to hard-code the specific formulations derived from them (e.g., (8) for the incremental encoding).…”

Section: Creating and Selecting Encodingsmentioning

confidence: 99%

Incremental and encoding formulations for Mixed Integer Programming

Yıldız

Vielma

2013

Operations Research Letters

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The standard way to represent a choice between n alternatives in Mixed Integer Programming is through n binary variables that add up to one. Unfortunately, this approach commonly leads to unbalanced branch-and-bound trees and diminished solver performance. In this paper, we present an encoding formulation framework that encompasses and expands existing approaches to mitigate this behavior. Through this framework, we generalize the incremental formulation for piecewise linear functions to any finite union of polyhedra with identical recession cones.

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Section: Creating and Selecting Encodingsmentioning

confidence: 99%

Incremental and encoding formulations for Mixed Integer Programming

Yıldız

Vielma

2013

Operations Research Letters

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“…The nonlinear functions in (6), (16), and (26) are linearized with the so-called extended incremental method [11]. The nonlinear and nonconvex set of (15) is linearized with outer approximation constraints [16].…”

Section: Linearized Formulationmentioning

confidence: 99%

“…1) The extended incremental method is used for the linearization of the gas physics [11]. This linearization method builds on the simple incremental method, exhibiting enhanced features and advantages as compared to standard piecewise linear approximation methods.…”

Section: Introductionmentioning

confidence: 99%

Linearized model for optimization of coupled electricity and natural gas systems

Sirvent

Kanelakis

GEIßLER

et al. 2017

J. Mod. Power Syst. Clean Energy

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In this paper a combined optimization of a coupled electricity and gas system is presented. For the electricity network a unit commitment problem with optimization of energy and reserves under a power pool, considering all system operational and unit technical constraints is solved. The gas network subproblem is a medium-scale mixed-integer nonconvex and nonlinear programming problem. The coupling constraints between the two networks are nonlinear as well. The resulting mixed-integer nonlinear program is linearized with the extended incremental method and an outer approximation technique. The resulting model is evaluated using the Greek power and gas system comprising fourteen gas-fired units under four different approximation accuracy levels. The results indicate the efficiency of the proposed mixedinteger linear program model and the interplay between computational requirements and accuracy.

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“…A common approach is to approximate the nonlinear functions with piecewise-linear functions, and then solve the corresponding approximation as a mixed-integer linear program (MILP) (e.g., [1,3,8,9,15,16,20,27,26]). In this section, we describe one MILP approximation and two MILP relaxations of the set MINLP2.…”

Section: Piecewise-linear Approximations and Relaxationsmentioning

confidence: 99%

Models and solution techniques for production planning problems with increasing byproducts

2014

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We consider a production planning problem where the production process creates a mixture of desirable products and undesirable byproducts. In this production process, at any point in time the fraction of the mixture that is an undesirable byproduct increases monotonically as a function of the cumulative mixture production up to that time. The mathematical formulation of this continuous-time problem is nonconvex.We present a discrete-time mixed-integer nonlinear programming (MINLP) formulation that exploits the increasing nature of the byproduct ratio function. We demonstrate that this new formulation is more accurate than a previously proposed MINLP formulation. We describe three different mixed-integer linear programming (MILP) approximation and relaxation models of this nonconvex MINLP, and derive modifications that strengthen the linear programming relaxations of these models. We also introduce nonlinear programming formulations to choose piecewise-linear approximations and relaxations of multiple functions that share the same domain and use the same set of break points in the domain.We conclude with computational experiments that demonstrate that the proposed formulation is more accurate than the previous formulation, and that the strengthened MILP approximation and relaxation models can be used to obtain provably near-optimal solutions for large instances of this nonconvex MINLP. Experiments also illustrate the quality of the piecewise-linear approximations produced by our nonlinear programming formulations.

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Using Piecewise Linear Functions for Solving MINLPs

Cited by 104 publications

References 24 publications

Incremental and encoding formulations for Mixed Integer Programming

Incremental and encoding formulations for Mixed Integer Programming

Linearized model for optimization of coupled electricity and natural gas systems

Models and solution techniques for production planning problems with increasing byproducts

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