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Discrete optimization methods to fit piecewise affine models to data points

Discrete optimization methods to fit piecewise affine models to data points
Discrete optimization methods to fit piecewise affine models to data points
Fitting piecewise affine models to data points is a pervasive task in many scientific disciplines. In this work, we address the k-Piecewise Affine Model Fitting with Piecewise Linear Separability problem (k-PAMF-PLS) where, given a set of m points {a1,…,am}?Rn{a1,…,am}?Rn and the corresponding observations {b1,…,bm}?R{b1,…,bm}?R, we have to partition the domain RnRn into k piecewise linearly (or affinely) separable subdomains and to determine an affine submodel (function) for each of them so as to minimize the total linear fitting error w.r.t. the observations bi.

To solve k-PAMF-PLS to optimality, we propose a mixed-integer linear programming (MILP) formulation where symmetries are broken by separating shifted column inequalities. For medium-to-large scale instances, we develop a four-step heuristic involving, among others, a point reassignment step based on the identification of critical points and a domain partition step based on multicategory linear classification. Differently from traditional approaches proposed in the literature for similar fitting problems, in both our exact and heuristic methods the domain partitioning and submodel fitting aspects are taken into account simultaneously.

Computational experiments on real-world and structured randomly generated instances show that, with our MILP formulation with symmetry breaking constraints, we can solve to proven optimality many small-size instances. Our four-step heuristic turns out to provide close-to-optimal solutions for small-size instances, while allowing to tackle instances of much larger size. The experiments also show that the combined impact of the main features of our heuristic is quite substantial when compared to standard variants not including them. We conclude with an application to the identification of dynamical piecewise affine systems for which we obtain promising results of comparable quality with those achieved with state-of-the-art methods from the literature on benchmark data sets.
0305-0548
214-230
Amaldi, Edoardo
eefad18b-86c1-4a8f-a931-23e8ddd59d6e
Coniglio, Stefano
03838248-2ce4-4dbc-a6f4-e010d6fdac67
Taccari, Leonardo
53ecb5ea-8f74-417e-aef9-ff31dbb33b2f
Amaldi, Edoardo
eefad18b-86c1-4a8f-a931-23e8ddd59d6e
Coniglio, Stefano
03838248-2ce4-4dbc-a6f4-e010d6fdac67
Taccari, Leonardo
53ecb5ea-8f74-417e-aef9-ff31dbb33b2f

Amaldi, Edoardo, Coniglio, Stefano and Taccari, Leonardo (2016) Discrete optimization methods to fit piecewise affine models to data points. Computers & Operations Research, 75, 214-230. (doi:10.1016/j.cor.2016.05.001).

Record type: Article

Abstract

Fitting piecewise affine models to data points is a pervasive task in many scientific disciplines. In this work, we address the k-Piecewise Affine Model Fitting with Piecewise Linear Separability problem (k-PAMF-PLS) where, given a set of m points {a1,…,am}?Rn{a1,…,am}?Rn and the corresponding observations {b1,…,bm}?R{b1,…,bm}?R, we have to partition the domain RnRn into k piecewise linearly (or affinely) separable subdomains and to determine an affine submodel (function) for each of them so as to minimize the total linear fitting error w.r.t. the observations bi.

To solve k-PAMF-PLS to optimality, we propose a mixed-integer linear programming (MILP) formulation where symmetries are broken by separating shifted column inequalities. For medium-to-large scale instances, we develop a four-step heuristic involving, among others, a point reassignment step based on the identification of critical points and a domain partition step based on multicategory linear classification. Differently from traditional approaches proposed in the literature for similar fitting problems, in both our exact and heuristic methods the domain partitioning and submodel fitting aspects are taken into account simultaneously.

Computational experiments on real-world and structured randomly generated instances show that, with our MILP formulation with symmetry breaking constraints, we can solve to proven optimality many small-size instances. Our four-step heuristic turns out to provide close-to-optimal solutions for small-size instances, while allowing to tackle instances of much larger size. The experiments also show that the combined impact of the main features of our heuristic is quite substantial when compared to standard variants not including them. We conclude with an application to the identification of dynamical piecewise affine systems for which we obtain promising results of comparable quality with those achieved with state-of-the-art methods from the literature on benchmark data sets.

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Accepted/In Press date: 1 May 2016
e-pub ahead of print date: 3 May 2016
Published date: November 2016
Organisations: Operational Research

Identifiers

Local EPrints ID: 394029
URI: http://eprints.soton.ac.uk/id/eprint/394029
ISSN: 0305-0548
PURE UUID: 8d1bdc45-9cd9-480e-8246-e7073ba2dd7d
ORCID for Stefano Coniglio: ORCID iD orcid.org/0000-0001-9568-4385

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Date deposited: 10 May 2016 13:16
Last modified: 17 Dec 2019 06:49

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