Topic
Piecewise linear function
About: Piecewise linear function is a research topic. Over the lifetime, 8133 publications have been published within this topic receiving 161444 citations.
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01 Jan 2007
TL;DR: In this paper, the authors extend the input-to-state stability (ISS) framework to continuous-time discontinuous dynamical systems (DDS) adopting piecewise smooth Lyapunov functions.
Abstract: In this paper we will extend the input-to-state stability (ISS) framework to continuous-time discontinuous dynamical systems (DDS) adopting piecewise smooth ISS Lyapunov functions. The main motivation for investigating piecewise smooth ISS Lyapunov functions is the success of piecewise smooth Lyapunov functions in the stability analysis of hybrid systems. This paper proposes an extension of the well-known Filippov's solution concept, that is appropriate for 'open' systems so as to allow interconnections of DDS. It is proven that the existence of a piecewise smooth ISS Lyapunov function for a DDS implies ISS. In addition, a (small gain) ISS interconnection theorem is derived for two DDS that both admit a piecewise smooth ISS Lyapunov function. This result is constructive in the sense that an explicit ISS Lyapunov function for the interconnected system is given. It is shown how these results can be applied to construct piecewise quadratic ISS Lyapunov functions for piecewise linear systems (including sliding motions) via linear matrix inequalities.
70 citations
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TL;DR: A hybrid meta-heuristic algorithm is designed to solve a routing problem in urban transportation which considers time-dependent travel time, multiple trips per vehicle, and loading time at the depot simultaneously, and is shown to be robust and efficient under different speed profiles and maximum trip duration limits.
70 citations
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TL;DR: In this article, a revised version of the CPA Lyapunov method was proposed to compute continuous and piecewise affine functions for systems with asymptotically stable equilibria.
70 citations
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TL;DR: In this article, strong mixed-integer programming (MIP) formulations for high-dimensional piecewise linear functions that correspond to trained neural networks are presented, which can be used for a number of important tasks such as verifying that an image classification network is robust to adversarial inputs, or solving decision problems where the objective function is a machine learning model.
Abstract: We present strong mixed-integer programming (MIP) formulations for high-dimensional piecewise linear functions that correspond to trained neural networks. These formulations can be used for a number of important tasks, such as verifying that an image classification network is robust to adversarial inputs, or solving decision problems where the objective function is a machine learning model. We present a generic framework, which may be of independent interest, that provides a way to construct sharp or ideal formulations for the maximum of d affine functions over arbitrary polyhedral input domains. We apply this result to derive MIP formulations for a number of the most popular nonlinear operations (e.g. ReLU and max pooling) that are strictly stronger than other approaches from the literature. We corroborate this computationally, showing that our formulations are able to offer substantial improvements in solve time on verification tasks for image classification networks.
70 citations
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04 Nov 2001TL;DR: This paper presents an approach to the nonlinear model reduction based on representing the non linear system with a piecewise-linear system and then reducing each of the pieces with a Krylov projection, and shows that the macromodels obtained are significantly more accurate than models obtained with linear or the recently developed quadratic reduction techniques.
Abstract: In this paper we present an approach to the nonlinear model reduction based on representing the nonlinear system with a piecewise-linear system and then reducing each of the pieces with a Krylov projection. However, rather than approximating the individual components as piecewise-linear and then composing hundreds of components to make a system with exponentially many different linear regions, we instead generate a small set of linearizations about the state trajectory which is the response to a 'training input'. Computational results and performance data are presented for a nonlinear circuit and a micromachined fixed-fixed beam example. These examples demonstrate that the macromodels obtained with the proposed reduction algorithm are significantly more accurate than models obtained with linear or the recently developed quadratic reduction techniques. Finally, it is shown that the proposed technique is computationally inexpensive, and that the models can be constructed 'on-the-fly', to accelerate simulation of the system response.
70 citations