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Mixed-Integer Programming Formulations for Piecewise Linear Functions

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Encyclopedia of Optimization


Piecewise linear (PWL) functions are comprised of a series of connected, affine segments which intersect at breakpoints. PWL functions can be used to model trends in data, such as the damage caused by post-fire debris flow [14]. A more common application of PWL functions is to model continuous, non-linear functions [4]. Therefore, mixed-integer non-linear programming (MINLP) problems can be approximately solved using only mixed-integer linear programming (MILP) techniques. When approximating convex functions, linear programming (LP) techniques can be used; however, with nonconvex functions, the optimisation problems are NP-hard even in the univariate case [12].

Many models have been presented in the literature to formulate PWL functions. This entry presents a discussion on SOS2, convex, disaggregated and incremental formulations, as well as recent advances for logarithmic formulations. Such formulations were first compiled into a unified framework by Vielma et al. in 2010 [

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Warwicker, J.A., Rebennack, S. (2023). Mixed-Integer Programming Formulations for Piecewise Linear Functions. In: Pardalos, P.M., Prokopyev, O.A. (eds) Encyclopedia of Optimization. Springer, Cham.

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