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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TL;DR: In this article, a piecewise linear hysteresis circuit is considered and a one-dimensional return map is defined and its analytic formula is derived, which enables us to give a sufficient condition for chaos generation and to analyse bifurcation phenomena rigorously.
Abstract: This paper considers a simple piecewise linear hysteresis circuit. We define one-dimensional return map and derive its analytic formula. It enables us to give a sufficient condition for chaos generation and to analyse bifurcation phenomena rigorously. Especially, we have discovered period-doubling bifurcation with symmetry breaking. Some of theoretical results are verified by laboratory measurements.
62 citations
TL;DR: The single machine problems are proved to be NP-hard, and some properties of their optimal solutions are established, and a pseudopolynomial time algorithm is constructed for makespan minimization.
Abstract: We study the problems of scheduling a set of nonpreemptive jobs on a single or multiple machines without idle times where the processing time of a job is a piecewise linear nonincreasing function of its start time. The objectives are the minimization of makespan and minimization of total job completion time. The single machine problems are proved to be NP-hard, and some properties of their optimal solutions are established. A pseudopolynomial time algorithm is constructed for makespan minimization. Several heuristics are derived for both total completion time and makespan minimization. Computational experiments are conducted to evaluate their efficiency. NP-hardness proofs and polynomial time algorithms are presented for some special cases of the parallel machine problems. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 531–554, 2003
62 citations
TL;DR: A piecewise linear equation is proposed as a method of analysis of mathematical models of neural networks using a symbolic representation of the dynamics in this equation as a directed graph on an N-dimensional hypercube.
Abstract: A piecewise linear equation is proposed as a method of analysis of mathematical models of neural networks. A symbolic representation of the dynamics in this equation is given as a directed graph on an N-dimensional hypercube. This provides a formal link with discrete neural networks such as the original Hopfield models. Analytic criteria are given to establish steady states and limit cycle oscillations independent of network dimension. Model networks that display multiple stable limit cycles and chaotic dynamics are discussed. The results show that such equations are a useful and efficient method of investigating the behavior of neural networks.
62 citations
TL;DR: In this paper, the authors present a methodology for robust optimization, which is a methodology that has gained a lot of attention in the recent years, mainly due to the simplicity of the modeling process and ease of resolution even for large scale.
Abstract: Robust optimization is a methodology that has gained a lot of attention in the recent years. This is mainly due to the simplicity of the modeling process and ease of resolution even for large scale...
62 citations
TL;DR: In this article, a general statistical mechanical analysis is presented for networks of graded-response neurons whose dynamics is described by a system of differential RC-charging equations, which requires that the dynamics is governed by a Lyapunov function, a condition that is met for networks whose synaptic matrix is symmetric, and whose neurons have monotonically increasing input-output relations may be arbitrary.
Abstract: A general statistical mechanical analysis is presented for networks of graded-response neurons whose dynamics is described by a system of differential RC-charging equations. The analysis requires that the dynamics is governed by a Lyapunov function, a condition that is met for networks whose synaptic matrix is symmetric, and whose neurons have monotonically increasing input-output relations may be arbitrary. In particular, they may vary from neuron to neuron. As examples, we study networks with synaptic couplings as in the Hopfield model: two homogeneous networks consisting of neurons with a sigmoidal or a piecewise linear input-output characteristics. Apart from this, the input-output relation, and a network containing a random mixture of these two neuron types.
61 citations