scholarly journals The Unit Re-Balancing Problem

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3205
Author(s):  
Robin Dee ◽  
Armin Fügenschuh ◽  
George Kaimakamis
Keyword(s):  
Numerical Solutions ◽  
Piecewise Linear ◽  
Geographic Area ◽  
Piecewise Linear Function ◽  
Linear Functions ◽  
Mixed Integer ◽  
Mixed Integer Linear Program ◽  
Commodity Flow ◽  
The Status ◽  
Status Assessment

We describe the problem of re-balancing a number of units distributed over a geographic area. Each unit consists of a number of components. A value between 0 and 1 describes the current rating of each component. By a piecewise linear function, this value is converted into a nominal status assessment. The lowest of the statuses determines the efficiency of a unit, and the highest status its cost. An unbalanced unit has a gap between these two. To re-balance the units, components can be transferred. The goal is to maximize the efficiency of all units. On a secondary level, the cost for the re-balancing should be minimal. We present a mixed-integer nonlinear programming formulation for this problem, which describes the potential movement of components as a multi-commodity flow. The piecewise linear functions needed to obtain the status values are reformulated using inequalities and binary variables. This results in a mixed-integer linear program, and numerical standard solvers are able to compute proven optimal solutions for instances with up to 100 units. We present numerical solutions for a set of open test instances and a bi-criteria objective function, and discuss the trade-off between cost and efficiency.

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.

scholarly journals A Combinatorial Approach for Small and Strong Formulations of Disjunctive Constraints

2019 ◽  
Vol 44 (3) ◽  
pp. 793-820 ◽  
Author(s):  
Joey Huchette ◽  
Juan Pablo Vielma
Keyword(s):  
Piecewise Linear ◽  
Expressive Power ◽  
Complete Characterization ◽  
Linear Functions ◽  
Mixed Integer ◽  
Ordered Sets ◽  
Special Ordered Sets ◽  
Disjunctive Constraints

A framework is presented for constructing strong mixed-integer programming formulations for logical disjunctive constraints. This approach is a generalization of the logarithmically sized formulations of Vielma and Nemhauser for special ordered sets of type 2 (SOS2) constraints, and a complete characterization of its expressive power is offered. The framework is applied to a variety of disjunctive constraints, producing novel small and strong formulations for outer approximations of multilinear terms, generalizations of special ordered sets, piecewise linear functions over a variety of domains, and obstacle avoidance constraints.

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.

UNIVERSAL CNN CELLS

1999 ◽  
Vol 09 (01) ◽  
pp. 1-48 ◽  
Author(s):  
RADU DOGARU ◽  
LEON O. CHUA
Keyword(s):  
Linear Function ◽  
Boolean Functions ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Linear Functions ◽  
Compact Representation ◽  
Value Functions ◽  
Linear Discriminant ◽  
Absolute Value ◽  
Parity Function

A cellular neural/nonlinear network (CNN) [Chua, 1998] is a biologically inspired system where computation emerges from a collection of simple nonlinear locally coupled cells. This paper reviews our recent research results beginning from the standard uncoupled CNN cell which can realize only linearly separable local Boolean functions, to a generalized universal CNN cell capable of realizing arbitrary Boolean functions. The key element in this evolutionary process is the replacement of the linear discriminant (offset) function w(σ)=σ in the "standard" CNN cell in [Chua, 1998] by a piecewise-linear function defined in terms of only absolute value functions. As in the case of the standard CNN cells, the excitation σ evaluates the correlation between a given input vector u formed by the outputs of the neighboring cells, and a template vector b, which is interpreted in this paper as an orientation vector. Using the theory of canonical piecewise-linear functions [Chua & Kang, 1977], the discriminant function [Formula: see text] is found to guarantee universality and its parameters can be easily determined. In this case, the number of additional parameters and absolute value functions m is bounded by m<2n-1, where n is the number of all inputs (n=9 for a 3×3 template). An even more compact representation where m<n-1 is also presented which is based on a special form of a piecewise-linear function; namely, a multi-nested discriminant: w (σ) =s (zm +| zm -1 +⋯ | z1 +| z0 +σ |||). Using this formula, the "benchmark" Parity function with an arbitrary number of inputs n is found to have an analytical solution with a complexity of only m =O ( log 2 (n)).

Approximation factor of the piecewise linear functions in Mamdani fuzzy system and its realization process1

2021 ◽  
pp. 1-15
Author(s):  
Yujie Tao ◽  
Chunfeng Suo ◽  
Guijun Wang
Keyword(s):  
Continuous Function ◽  
Linear Function ◽  
Hypothesis Test ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Two Dimensions ◽  
Linear Functions ◽  
Piecewise Linear Functions ◽  
Approximation Accuracy ◽  
Approximation Factor

Piecewise linear function (PLF) is not only a generalization of univariate segmented linear function in multivariate case, but also an important bridge to study the approximation of continuous function by Mamdani and Takagi-Sugeno fuzzy systems. In this paper, the definitions of the PLF and subdivision are introduced in the hyperplane, the analytic expression of PLF is given by using matrix determinant, and the concept of approximation factor is first proposed by using m-mesh subdivision. Secondly, the vertex coordinates and their changing rules of the n-dimensional small polyhedron are found by dividing a three-dimensional cube, and the algebraic cofactor and matrix norm of corresponding determinants of piecewise linear functions are given. Finally, according to the method of solving algebraic cofactors and matrix norms, it is proved that the approximation factor has nothing to do with the number of subdivisions, but the approximation accuracy has something to do with the number of subdivisions. Furthermore, the process of a specific binary piecewise linear function approaching a continuous function according to infinite norm in two dimensions space is realized by a practical example, and the validity of PLFs to approximate a continuous function is verified by t-hypothesis test in Statistics.

scholarly journals Security-Constrained Unit Commitment Based on a Realizable Energy Delivery Formulation

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Hongyu Wu ◽  
Qiaozhu Zhai ◽  
Xiaohong Guan ◽  
Feng Gao ◽  
Hongxing Ye
Keyword(s):  
Unit Commitment ◽  
Piecewise Linear ◽  
Optimal Solution ◽  
Piecewise Linear Function ◽  
Mixed Integer ◽  
Comparative Case Study ◽  
Dynamic Programming Method ◽  
Individual Unit ◽  
New Formulation ◽  
Bus System

Security-constrained unit commitment (SCUC) is an important tool for independent system operators in the day-ahead electric power market. A serious issue arises that the energy realizability of the staircase generation schedules obtained in traditional SCUC cannot be guaranteed. This paper focuses on addressing this issue, and the basic idea is to formulate the power output of thermal units as piecewise-linear function. All individual unit constraints and systemwide constraints are then reformulated. The new SCUC formulation is solved within the Lagrangian relaxation (LR) framework, in which a double dynamic programming method is developed to solve individual unit subproblems. Numerical testing is performed for a 6-bus system and an IEEE 118-bus system on Microsoft Visual C# .NET platform. It is shown that the energy realizability of generation schedules obtained from the new formulation is guaranteed. Comparative case study is conducted between LR and mixed integer linear programming (MILP) in solving the new formulation. Numerical results show that the near-optimal solution can be obtained efficiently by the proposed LR-based method.

scholarly journals Models and Algorithms for Optimal Piecewise-Linear Function Approximation

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Eduardo Camponogara ◽  
Luiz Fernando Nazari
Keyword(s):  
Function Approximation ◽  
Linear Models ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Linear Functions ◽  
Piecewise Linear Functions ◽  
Linear Function Approximation ◽  
Data Points ◽  
Stochastic Functions ◽  
Piecewise Linear Models

Piecewise-linear functions can approximate nonlinear and unknown functions for which only sample points are available. This paper presents a range of piecewise-linear models and algorithms to aid engineers to find an approximation that fits best their applications. The models include piecewise-linear functions with a fixed and maximum number of linear segments, lower and upper envelopes, strategies to ensure continuity, and a generalization of these models for stochastic functions whose data points are random variables. Derived from recursive formulations, the algorithms are applied to the approximation of the production function of gas-lifted oil wells.

scholarly journals Empirical Bounds on Linear Regions of Deep Rectifier Networks

2020 ◽  
Vol 34 (04) ◽  
pp. 5628-5635 ◽  
Author(s):  
Thiago Serra ◽  
Srikumar Ramalingam
Keyword(s):  
Mixed Integer Linear Programming ◽  
Deep Neural Network ◽  
Piecewise Linear ◽  
Linear Functions ◽  
Mixed Integer ◽  
Piecewise Linear Functions ◽  
Activation Patterns ◽  
Solution Sets ◽  
Exact Counting ◽  
Trained Network

We can compare the expressiveness of neural networks that use rectified linear units (ReLUs) by the number of linear regions, which reflect the number of pieces of the piecewise linear functions modeled by such networks. However, enumerating these regions is prohibitive and the known analytical bounds are identical for networks with same dimensions. In this work, we approximate the number of linear regions through empirical bounds based on features of the trained network and probabilistic inference. Our first contribution is a method to sample the activation patterns defined by ReLUs using universal hash functions. This method is based on a Mixed-Integer Linear Programming (MILP) formulation of the network and an algorithm for probabilistic lower bounds of MILP solution sets that we call MIPBound, which is considerably faster than exact counting and reaches values in similar orders of magnitude. Our second contribution is a tighter activation-based bound for the maximum number of linear regions, which is particularly stronger in networks with narrow layers. Combined, these bounds yield a fast proxy for the number of linear regions of a deep neural network.

Single Allocation Hub Location with Heterogeneous Economies of Scale

2021 ◽  
Author(s):  
Borzou Rostami ◽  
Masoud Chitsaz ◽  
Okan Arslan ◽  
Gilbert Laporte ◽  
Andrea Lodi
Keyword(s):  
Large Scale ◽  
Economies Of Scale ◽  
Piecewise Linear ◽  
Distance Matrix ◽  
Linear Functions ◽  
Mixed Integer ◽  
Hub Location ◽  
Solution Methods ◽  
Problem Instances ◽  
Single Allocation

The economies of scale in hub location is usually modeled by a constant parameter, which captures the benefits companies obtain through consolidation. In their article “Single allocation hub location with heterogeneous economies of scale,” Rostami et al. relax this assumption and consider hub-hub connection costs as piecewise linear functions of the flow amounts. This spoils the triangular inequality property of the distance matrix, making the classical flow-based model invalid and further complicates the problem. The authors tackle the challenge by building a mixed-integer quadratically constrained program and by developing a methodology based on constructing Lagrangian function, linear dual functions, and specialized polynomial-time algorithms to generate enhanced cuts. The developed method offers a new strategy in Benders-type decomposition through relaxing a set of complicating constraints in subproblems when such relaxation is tight. The results confirm the efficacy of the solution methods in solving large-scale problem instances.

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