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

Joint Assignment of Power, Routing, and Spectrum in Static Flexible-Grid Networks

This paper proposes a novel network planning strategy to jointly allocate physical layer resources together with the routing and spectrum assignment in transparent nonlinear flexible-grid optical networks with static traffic demands. The physical layer resources, such as power spectral density, modulation format, and carrier frequency, are optimized for each connection. By linearizing the Gaussian noise model, both an optimal formulation and a low complexity decomposition heuristic are proposed. Our methods minimize the spectrum usage of networks, while satisfying requirements on the throughput and quality of transmission. Compared with existing schemes that allocate a uniform power spectral density to all connections, our proposed methods relax this constraint and, thus, utilize network resources more efficiently. Numerical results show that by optimizing the power spectral density per connection, the spectrum usage can be reduced by around 20% over uniform power spectral density schemes