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Locally ideal formulations for piecewise linear functions with indicator variables

Srikrishna Sridhar, Jeff Linderoth, James Luedtke
2013 Operations Research Letters  
In this paper, we consider mixed integer linear programming (MIP) formulations for piecewise linear functions (PLFs) that are evaluated when an indicator variable is turned on.  ...  We describe modifications to standard MIP formulations for PLFs with desirable theoretical properties and superior computational performance in this context.  ...  PLFs where a binary indicator variable determines if the function is required to be evaluated.  ... 
doi:10.1016/j.orl.2013.08.010 fatcat:uodsz2p3svhktlelxu4q2prycq

A Review of Piecewise Linearization Methods

Ming-Hua Lin, John Gunnar Carlsson, Dongdong Ge, Jianming Shi, Jung-Fa Tsai
2013 Mathematical Problems in Engineering  
This study therefore provides a review of piecewise linearization methods and analyzes the computational efficiency of various piecewise linearization methods.  ...  for obtaining an approximated global optimal solution.  ...  He called such formulations locally ideal.  ... 
doi:10.1155/2013/101376 fatcat:zqbkl6b2ibdebb4bkoddiknccu

Models for representing piecewise linear cost functions

Ahmet B. Keha, Ismael R. de Farias, George L. Nemhauser
2004 Operations Research Letters  
We study formulations of linear programs with piecewise linear objective functions with and without additional binary variables.  ...  We show that the two formulations without additional binary variables have the same LP bounds as those of the corresponding formulations with binary variables and therefore are preferable for efficient  ...  Acknowledgments We are grateful to Manfred Padberg for providing comments on an earlier draft of the paper. This research was partially supported by NSF under grants DMI-0100020 and DMI-0121495.  ... 
doi:10.1016/s0167-6377(03)00059-2 fatcat:6mxpkmuvcbcbla25b5xaw72cy4

Alternative linearisations for the operating cost function of UC problems

Zhaowei Geng, Antonio J. Conejo, Qing Xia
2017 IET Generation, Transmission & Distribution  
The authors describe two efficient piecewise-linear formulations of the objective function of an UC problem that are not found in the power literature.  ...  They show the advantage of the proposed formulations for both deterministic and stochastic UC problems.  ...  Xia is partly supported by the State Grid Corporation of China, the project of "the theory and application of a novel power flow calculation approach with reduced nonlinearity".  ... 
doi:10.1049/iet-gtd.2016.1395 fatcat:c57psqhudbfh3gtcdkga74iwk4

Mixed Integer Linear Programming Formulation Techniques

Juan Pablo Vielma
2015 SIAM Review  
In this survey we review advanced MIP formulation techniques that result in stronger and/or smaller formulations for a wide class of problems.  ...  A wide range of problems can be modeled as Mixed Integer Linear Programming (MIP) problems using standard formulation techniques.  ...  a piecewise linear function f defined by (3.3) for polytopes Q i k i=1 that satisfy a special ordering condition, formulation (8.3) reduces to a standard MIP formulation for piecewise linear functions  ... 
doi:10.1137/130915303 fatcat:qhscwiolrncmrfjjdjo74ae6qe

Merits of the Incremental Method for modeling Piecewise Linear functions [article]

Mutaz Tuffaha, Jan Tommy Gravdahl
2018 arXiv   pre-print
Further, we propose a tighter formulation for optimization problems over separable univariate PWL functions with binary indicators by using the incremental method.  ...  Several techniques were proposed to model the Piecewise linear (PWL) functions, including convex combination, incremental and multiple choice methods.  ...  CONCLUSION The incremental method to model the PWL function is proved to give locally ideal formulations just like the SOS2 method.  ... 
arXiv:1802.04005v1 fatcat:vw5kkaeoezeg5duycwjkbbruo4

Computing tight bounds via piecewise linear functions through the example of circle cutting problems

Steffen Rebennack
2016 Mathematical Methods of Operations Research  
Piecewise Linear Functions and Circle Cutting 21 Table 5 Summary of formulations and locally idealness.  ...  The literature contains several different formulations for piecewise linear functions.  ... 
doi:10.1007/s00186-016-0546-0 fatcat:qtiw6syvi5dcrph2thkhcmmgku

Models and solution techniques for production planning problems with increasing byproducts

Srikrishna Sridhar, Jeffrey Linderoth, James Luedtke
2014 Journal of Global Optimization  
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  ...  Experiments also illustrate the quality of the piecewise-linear approximations produced by our nonlinear programming formulations.  ...  Wright for several suggestions that helped improve this work, and we thank Ignacio Grossmann for bringing the reference [12] to our attention.  ... 
doi:10.1007/s10898-014-0167-1 fatcat:asb5k5ztnvdlpdeut2m6gjqvgm

Structural Investigation of Piecewise Linearized Network Flow Problems

Frauke Liers, Maximilian Merkert
2016 SIAM Journal on Optimization  
Several methods for modeling piecewise linear functions which provide a complete description for a single network arc are known.  ...  We show how to strengthen the formulation for specific substructures consisting of multiple arcs by linear inequalities.  ...  In particular, we thank Thomas Lange for technical support. We are also indepted to the anonymous reviewers for their constructive comments on this paper.  ... 
doi:10.1137/15m1006751 fatcat:dp4v7lhuw5hklgta3nzt6bbhrm

A Graduated Nonconvex Regularization for Sparse High Dimensional Model Estimation

Thomas F. Coleman, Yuying Li
2014 Journal of Computer and Communications  
Many high dimensional data mining problems can be formulated as minimizing an empirical loss function with a penalty proportional to the number of variables required to describe a model.  ...  We prove that under some conditions the proposed regularization problem using the continuous piecewise linear approximation is equivalent to the original 0 l regularization problem.  ...  Selecting a model with a small number variables can be formulated as minimizing an empirical loss function with a penalization for the number of nonzero variables; this is referred to as 0 l -regularization  ... 
doi:10.4236/jcc.2014.211001 fatcat:a4633lxlufcnhnn3dl2pgxvs6y

Integrating linear physical programming within collaborative optimization for multiobjective multidisciplinary design optimization

C.D. McAllister, T.W. Simpson, K. Hacker, K. Lewis, A. Messac
2004 Structural And Multidisciplinary Optimization  
Multidisciplinary design optimization (MDO) is a concurrent engineering design tool for large-scale, complex systems design that can be affected through the optimal design of several smaller functional  ...  It is solved using the proposed formulation and compared against a traditional formulation without collaborative optimization or linear physical programming.  ...  single-level goal programming formulation with a piecewise linear preference function for each design criterion.  ... 
doi:10.1007/s00158-004-0481-1 fatcat:3cbtrjwzendwbnhgl3tfgirilq

Page 7562 of Mathematical Reviews Vol. , Issue 2002J [page]

2002 Mathematical Reviews  
It is shown, for linear kinematic and isotropic hardening, that this method represents a piecewise smooth mapping as well.  ...  The reformulation is done using the minimum function. The system of equations obtained is piecewise smooth which en- ables Pang’s Newton method for B-differentiable equations to be used.  ... 

Between Mathematical Programming and Systems Theory: Linear Complementarity Systems [chapter]

Hans Schumacher
2001 Systems & Control: Foundations & Applications  
A solution concept for linear complementarity systems is provided, and conditions for existence and uniqueness of solutions are given.  ...  Complementarity systems arise from the interconnection of an input-output system (of the type well known in mathematical systems theory) with a set of complementarity conditions (of the type well known  ...  To illustrate this, consider the specific case of a linear passive system coupled with a number of ideal diodes.  ... 
doi:10.1007/978-1-4612-0179-3_11 fatcat:whgx6azyxnfhzbkas7xfn6evki

An adaptive strategy based on conforming quadtree meshes for kinematic limit analysis

H. Nguyen-Xuan, Hien V. Do, Khanh N. Chau
2018 Computer Methods in Applied Mechanics and Engineering  
Piecewise-linear shape functions enhanced with generalized bubble functions in barycentric coordinates are used to approximate the velocity field.  ...  However, a major difficulty in finite element formulations is the appearance of hanging nodes during adaptive process.  ...  Son Nguyen-Hoang for his comments and suggestions to this work.  ... 
doi:10.1016/j.cma.2018.06.023 fatcat:bwiyh4t7bjcppkpbbs4z76ukfm

Expressing Special Structures in an Algebraic Modeling Language for Mathematical Programming

Robert Fourer, David M. Gay
1995 Orsa Journal On Computing  
As a step in this direction, we describe extensions to an algebraic modeling language that encompass piecewise-linear, network and related structures.  ...  A knowledge of the presence of certain special structures can be advantageous in both the formulation and solution of linear programming problems.  ...  The piecewise-linear version has 3000 constraints in 3025 variables, with 4 or 5 linear pieces per variable.  ... 
doi:10.1287/ijoc.7.2.166 fatcat:ueb4xbt6v5fepl5kv56gheuc3a
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