LinA: A faster approach to piecewise linear approximations using corridors and its application to mixed-integer optimization
Résumé
In this paper, we address the problem of approximating and over/under-estimating univariate functions with piecewise linear (PWL) functions with the minimum number of linear segments given a bound on the pointwise approximation error allowed. Through a new geometric approach and building on the work of Ngueveu [Ngu19], we develop new algorithms that can solve the problem in quasi-logarithmic time on a very broad class of error types. Such algorithms find many applications, mostly related to solving certain classes of (mixed-integer) nonlinear and nonconvex programming (MINLP) problems by mixed-integer linear programming (MILP) techniques. An efficient implementation of our algorithms is available as a Julia package. Benchmarks are also provided to showcase how our method outperforms the state-of-the-art for this problem. Finally, we show how our algorithms can be used to efficiently solve certain classes of MINLP problems by a case study on multicommodity network design problems with congestion.
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