piecewise linear
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2022 ◽  
Vol 23 (1) ◽  
pp. 1-35
Manuel Bodirsky ◽  
Marcello Mamino ◽  
Caterina Viola

Valued constraint satisfaction problems (VCSPs) are a large class of combinatorial optimisation problems. The computational complexity of VCSPs depends on the set of allowed cost functions in the input. Recently, the computational complexity of all VCSPs for finite sets of cost functions over finite domains has been classified. Many natural optimisation problems, however, cannot be formulated as VCSPs over a finite domain. We initiate the systematic investigation of the complexity of infinite-domain VCSPs with piecewise linear homogeneous cost functions. Such VCSPs can be solved in polynomial time if the cost functions are improved by fully symmetric fractional operations of all arities. We show this by reducing the problem to a finite-domain VCSP which can be solved using the basic linear program relaxation. It follows that VCSPs for submodular PLH cost functions can be solved in polynomial time; in fact, we show that submodular PLH functions form a maximally tractable class of PLH cost functions.

Morteza Kimiaei ◽  
Arnold Neumaier ◽  
Behzad Azmi

AbstractRecently, Neumaier and Azmi gave a comprehensive convergence theory for a generic algorithm for bound constrained optimization problems with a continuously differentiable objective function. The algorithm combines an active set strategy with a gradient-free line search along a piecewise linear search path defined by directions chosen to reduce zigzagging. This paper describes , an efficient implementation of this scheme. It employs new limited memory techniques for computing the search directions, improves by adding various safeguards relevant when finite precision arithmetic is used, and adds many practical enhancements in other details. The paper compares and several other solvers on the unconstrained and bound constrained problems from the collection and makes recommendations on which solver to use and when. Depending on the problem class, the problem dimension, and the precise goal, the best solvers are , , and .

2022 ◽  
pp. 1-18
Yuanfeng Xia ◽  
Jian Pang

Abstract The transient vibro-impacts induced by clearance between the connected rotors in driveline system easily causes serious transient noise and vibration, especially between the gear teeth with backlash. To analyze the transient vibro-impacts of the driveline system excited by a step-down engine torque, a new piecewise nonlinear clearance element with time-varying stiffness and oil squeeze damping is proposed, and an 8 degree-of-freedom lumped parameters model with the new piecewise nonlinear clearance elements is established. The transient vibro-impact phenomena of the vehicle driveline during fast disengagement of the clutch are numerically simulated. Colormaps of angular acceleration and vibro-impact force shows the difference of frequency components from transient impact to stable tooth-meshing. The phase plane reveals the phenomenon of multiple impacts and rebounds in each transient impact, and shows the relationship between the relative contact displacement and velocity. The frequency responses of the angular velocity, angular acceleration and vibro-impact forces with time-varying stiffness and linear stiffness are compared respectively. Compared with the widely used clearance element with piecewise linear stiffness, the new nonlinear clearance element with the piecewise nonlinear time-varying stiffness can better reveal the transient vibro-impact responses between the driving and driven gears. Lastly, the transient vibro-impact results of driveline system are verified by the vehicle experiments.

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 198
Loay Alkhalifa ◽  
Hans Mittelmann

Techniques and methods of linear optimization underwent a significant improvement in the 20th century which led to the development of reliable mixed integer linear programming (MILP) solvers. It would be useful if these solvers could handle mixed integer nonlinear programming (MINLP) problems. Piecewise linear approximation (PLA) is one of most popular methods used to transform nonlinear problems into linear ones. This paper will introduce PLA with brief a background and literature review, followed by describing our contribution before presenting the results of computational experiments and our findings. The goals of this paper are (a) improving PLA models by using nonuniform domain partitioning, and (b) proposing an idea of applying PLA partially on MINLP problems, making them easier to handle. The computational experiments were done using quadratically constrained quadratic programming (QCQP) and MIQCQP and they showed that problems under PLA with nonuniform partition resulted in more accurate solutions and required less time compared to PLA with uniform partition.

Chao Zhang ◽  
Wen Wang ◽  
Pan Yong ◽  
Lina Cheng ◽  
Shoupei Zhai ◽  

Abstract Baseline drift caused by slowly changing environment and other instability factors affects significantly the performance of gas sensors, resulting in reduced accuracy of gas classification and quantification of the electronic nose. In this work, a two-stage method is proposed for real-time sensor baseline drift compensation based on estimation theory and piecewise linear approximation. In the first stage, the linear information from the baseline before exposure is extracted for prediction. The second stage continuously predicts changing linear parameters during exposure by combining temperature change information and time series information, and then the baseline drift is compensated by subtracting the predicted baseline from the real sensor response. The proposed method is compared to three efficient algorithms and the experiments are conducted towards two simulated datasets and two surface acoustic wave sensor datasets. The experimental results prove the effectiveness of the proposed algorithm. Moreover, the proposed method can recover the true response signal under different ambient temperatures in real-time, which can guide the future design of low-power and low-cost rapid detection systems.

2022 ◽  
Arman Kheirati Roonizi

<pre>$\ell_2$ and $\ell_1$ trend filtering are two of the most popular denoising algorithms that are widely used in science, engineering, and statistical signal and image processing applications. They are typically treated as separate entities, with the former as a linear time invariant (LTI) filter which is commonly used for smoothing the noisy data and detrending the time-series signals while the latter is a nonlinear filtering method suited for the estimation of piecewise-polynomial signals (\eg, piecewise-constant, piecewise-linear, piecewise-quadratic and \etc) observed in additive white Gaussian noise. In this article, we propose a Kalman filtering approach to design and implement $\ell_2$ and $\ell_1$ trend filtering % (QV and TV regularization) with the aim of teaching these two approaches and explaining their differences and similarities. Hopefully the framework presented in this article will provide a straightforward and unifying platform for understanding the basis of these two approaches. In addition, the material may be useful in lecture courses in signal and image processing, or indeed, it could be useful to introduce our colleagues in signal processing to the application of Kalman filtering in the design of $\ell_2$ and $\ell_1$ trend filtering.</pre>

Christian Füllner ◽  
Steffen Rebennack

AbstractWe propose a new decomposition method to solve multistage non-convex mixed-integer (stochastic) nonlinear programming problems (MINLPs). We call this algorithm non-convex nested Benders decomposition (NC-NBD). NC-NBD is based on solving dynamically improved mixed-integer linear outer approximations of the MINLP, obtained by piecewise linear relaxations of nonlinear functions. Those MILPs are solved to global optimality using an enhancement of nested Benders decomposition, in which regularization, dynamically refined binary approximations of the state variables and Lagrangian cut techniques are combined to generate Lipschitz continuous non-convex approximations of the value functions. Those approximations are then used to decide whether the approximating MILP has to be dynamically refined and in order to compute feasible solutions for the original MINLP. We prove that NC-NBD converges to an $$\varepsilon $$ ε -optimal solution in a finite number of steps. We provide promising computational results for some unit commitment problems of moderate size.

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