scholarly journals MINLP formulations for continuous piecewise linear function fitting

2021 ◽  
Author(s):  
Noam Goldberg ◽  
Steffen Rebennack ◽  
Youngdae Kim ◽  
Vitaliy Krasko ◽  
Sven Leyffer
Keyword(s):  
Linear Function ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Mixed Integer ◽  
Function Fitting ◽  
Computational Performance ◽  
Integer Nonlinear Programming ◽  
Minlp Model ◽  
Necessary Constraints ◽  
Continuous Piecewise Linear Function

AbstractWe consider a nonconvex mixed-integer nonlinear programming (MINLP) model proposed by Goldberg et al. (Comput Optim Appl 58:523–541, 2014. 10.1007/s10589-014-9647-y) for piecewise linear function fitting. We show that this MINLP model is incomplete and can result in a piecewise linear curve that is not the graph of a function, because it misses a set of necessary constraints. We provide two counterexamples to illustrate this effect, and propose three alternative models that correct this behavior. We investigate the theoretical relationship between these models and evaluate their computational performance.

A Comparison of Two Mixed-Integer Linear Programs for Piecewise Linear Function Fitting

2021 ◽  
Author(s):  
John Alasdair Warwicker ◽  
Steffen Rebennack
Keyword(s):  
Linear Programming ◽  
Linear Function ◽  
Integer Linear Programming ◽  
Mixed Integer Linear Programming ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Mixed Integer ◽  
Data Sets ◽  
Binary Variables ◽  
Function Fitting

The problem of fitting continuous piecewise linear (PWL) functions to discrete data has applications in pattern recognition and engineering, amongst many other fields. To find an optimal PWL function, the positioning of the breakpoints connecting adjacent linear segments must not be constrained and should be allowed to be placed freely. Although the univariate PWL fitting problem has often been approached from a global optimisation perspective, recently, two mixed-integer linear programming approaches have been presented that solve for optimal PWL functions. In this paper, we compare the two approaches: the first was presented by Rebennack and Krasko [Rebennack S, Krasko V (2020) Piecewise linear function fitting via mixed-integer linear programming. INFORMS J. Comput. 32(2):507–530] and the second by Kong and Maravelias [Kong L, Maravelias CT (2020) On the derivation of continuous piecewise linear approximating functions. INFORMS J. Comput. 32(3):531–546]. Both formulations are similar in that they use binary variables and logical implications modelled by big-[Formula: see text] constructs to ensure the continuity of the PWL function, yet the former model uses fewer binary variables. We present experimental results comparing the time taken to find optimal PWL functions with differing numbers of breakpoints across 10 data sets for three different objective functions. Although neither of the two formulations is superior on all data sets, the presented computational results suggest that the formulation presented by Rebennack and Krasko is faster. This might be explained by the fact that it contains fewer complicating binary variables and sparser constraints. Summary of Contribution: This paper presents a comparison of the mixed-integer linear programming models presented in two recent studies published in the INFORMS Journal on Computing. Because of the similarity of the formulations of the two models, it is not clear which one is preferable. We present a detailed comparison of the two formulations, including a series of comparative experimental results across 10 data sets that appeared across both papers. We hope that our results will allow readers to take an objective view as to which implementation they should use.

scholarly journals REGION INTERVISIBILITY IN TERRAINS

2007 ◽  
Vol 17 (04) ◽  
pp. 331-347
Author(s):  
ESTHER MOET ◽  
MARC VAN KREVELD ◽  
RENÉ VAN OOSTRUM
Keyword(s):  
Linear Function ◽  
Piecewise Linear ◽  
Time Algorithm ◽  
Piecewise Linear Function ◽  
Line Of Sight ◽  
Continuous Piecewise Linear Function ◽  
Simply Connected ◽  
And Storage

A polyhedral terrain is the graph of a continuous piecewise linear function defined over the triangles of a triangulation in the xy-plane. Two points on or above a terrain are visible to each other if the line-of-sight does not intersect the space below the terrain. In this paper, we look at three related visibility problems in terrains. Suppose we are given a terrain T with n triangles and two regions R1 and R2 on T, i.e., two simply connected subsets of at most m triangles. First, we present an algorithm that determines, for any constant ∊ > 0, within O(n1+∊m) time and storage whether or not R1 and R2 are completely intervisible. We also give an O(m3n4) time algorithm to determine whether every point in R1 sees at least one point in R2. Finally, we present an O(m2n2 log n) time algorithm to determine whether there exists a pair of points p ∈ R1 and q ∈ R2, such that p and q see each other.

scholarly journals DESIGN OF TIME DELAYED CHAOTIC CIRCUIT WITH THRESHOLD CONTROLLER

2011 ◽  
Vol 21 (03) ◽  
pp. 725-735 ◽  
Author(s):  
K. SRINIVASAN ◽  
I. RAJA MOHAMED ◽  
K. MURALI ◽  
M. LAKSHMANAN ◽  
SUDESHNA SINHA
Keyword(s):  
Numerical Simulations ◽  
Linear Function ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Operational Amplifiers ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Chaotic Oscillator ◽  
Threshold Values ◽  
Chaotic Circuit

A novel time delayed chaotic oscillator exhibiting mono- and double scroll complex chaotic attractors is designed. This circuit consists of only a few operational amplifiers and diodes and employs a threshold controller for flexibility. It efficiently implements a piecewise linear function. The control of piecewise linear function facilitates controlling the shape of the attractors. This is demonstrated by constructing the phase portraits of the attractors through numerical simulations and hardware experiments. Based on these studies, we find that this circuit can produce multi-scroll chaotic attractors by just introducing more number of threshold values.

Recognition of handwritten digits using a neural network based on the piecewise linear function

1994 ◽  
Author(s):  
Hongyu Liu ◽  
Yialei Wang ◽  
Peimao Sun ◽  
Tianyun Zhang ◽  
Rong Jiang
Keyword(s):  
Neural Network ◽  
Linear Function ◽  
Piecewise Linear ◽  
Piecewise Linear Function

Study on the Scheduling Problem of Powder Painting Production - A Case Study

2010 ◽  
Vol 97-101 ◽  
pp. 2459-2464
Author(s):  
Zhang Yong Hu ◽  
Qiang Su ◽  
Jun Liu ◽  
Hai Xia Yang
Keyword(s):  
Large Scale ◽  
Production Efficiency ◽  
Production Performance ◽  
Mixed Integer ◽  
Scheduling Problem ◽  
Adaptive Search ◽  
Optimal Sequence ◽  
Integer Nonlinear Programming ◽  
Minlp Model

A large-scale powder-painting scheduling problem is explored. The purpose is to find out the optimal sequence of a number of batches that dynamically arrive from upstream processes within a given scheduling horizon. The objective is to enhance the production efficiency and decrease the production cost as well. To solve this problem, a mixed integer nonlinear programming (MINLP) model is constructed and an algorithm called greedy randomized adaptive search procedure (GRASP) is designed. Case studies demonstrate that the proposed approach can improve the production performance significantly.

MAXIMUM DYNAMIC RANGE OF BIFURCATION OF CHUA'S CIRCUIT

1993 ◽  
Vol 03 (02) ◽  
pp. 471-481 ◽  
Author(s):  
A. A. A. NASSER ◽  
E. E. HOSNY ◽  
M. I. SOBHY
Keyword(s):  
Linear Function ◽  
Dynamic Range ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Bifurcation Parameter ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Simulation Results

This paper includes a method for detecting the maximum possible range of bifurcations based upon the multilevel oscillation technique. An application of the method to Chua's circuit, and new simulation results using the slope of the piecewise-linear function as a bifurcation parameter are presented.

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