Piecewise smooth extreme functions are piecewise linear

For the one dimensional infinite group relaxation, we construct a sequence of extreme valid functions that are piecewise linear and such that for every natural number $$k\ge 2$$, there is a function in the sequence with k slopes. This settles an open question in this area regarding a universal bound on the number of slopes for extreme functions. The function which is the pointwise limit of this sequence is an extreme valid function that is continuous and has an infinite number of slopes. This provides a new and more refined counterexample to an old conjecture of Gomory and Johnson that stated all extreme functions are piecewise linear.

Infinite group relaxations of integer programs (IP) were introduced by Gomory and Johnson (Math Program 3:23–85, 1972) to generate cutting planes for general IPs. These valid inequalities correspond to real-valued functions defined over an appropriate infinite group. Among all the valid inequalities of the infinite group relaxation, extreme inequalities are most important since they are the strongest cutting planes that can be obtained within the group-theoretic framework. However, very few properties of extreme inequalities of infinite group relaxations are known. In particular, it is not known if all extreme inequalities are continuous and what their relations are to extreme inequalities of finite group problems. In this paper, we describe new properties of extreme functions of infinite group problems. In particular, we study the behavior of the pointwise limit of a converging sequence of extreme functions as well as the relations between extreme functions of finite and infinite group problems. Using these results, we prove for the first time that a large class of discontinuous functions is extreme for infinite group problems. This class of extreme functions is the generalization of the functions given by Letchford and Lodi (Oper Res Lett 30(2):74–82, 2002), Dash and Gunluk (Proceedings 10th conference on integer programming and combinatorial optimization. Springer, Heidelberg, pp 33–45 (2004), Math Program 106:29–53, 2006) and Richard et al. (Math Program 2008, to appear). We also present several other new classes of discontinuous extreme functions. Surprisingly, we prove that the functions defining extreme inequalities for infinite group relaxations of mixed integer programs are continuous.

This is a survey on the infinite group problem, an infinite-dimensional relaxation of integer linear optimization problems introduced by Ralph Gomory and Ellis Johnson in their groundbreaking papers titled "Some continuous functions related to corner polyhedra I, II" [Math. Programming 3 (1972), 23-85, 359-389]. The survey presents the infinite group problem in the modern context of cut generating functions. It focuses on the recent developments, such as algorithms for testing extremality and breakthroughs for the k-row problem for general k >= 1 that extend previous work on the single-row and two-row problems. The survey also includes some previously unpublished results; among other things, it unveils piecewise linear extreme functions with more than four different slopes. An interactive companion program, implemented in the open-source computer algebra package Sage, provides an updated compendium of known extreme functions.

The moments produced by the aerodynamic control effectors on aircraft are typically non-linear functions of displacement with a region of linearity about zero displacement. In this paper we briefly review a novel approach that was developed at AFRL for solving the control allocation problem where the control moments are piecewise linear functions of the control displacement. The advantages of this approach are that it avoids cumbersome, high-order polynomial curve fits of the aerodynamic data and thus avoids solution of a non-linear programming problem. Stability and control derivatives are almost always stored in multi-dimensional look-up tables where it is assumed that the data is piecewise linear. The approach that is presented utilizes this piecewise linear assumption for the control effector moment data and accounts for non-linearities in the moment-effector relationships and position constraints on the effectors as well. The control allocation problem using piecewise linear functions is posed as a mixed-integer linear program (MILP) and solved using a freely available GNU-licensed mixed-integer linear programming code. We demonstrate two applications of the piecewise linear control allocation (PLCA) approach. The first application to be considered is to use the PLCA approach in the inner-loop control law of a re-usable launch vehicle on approach and landing. The second application is constraint estimation for trajectory reshaping and re-targeting. In order to successfully re-target a trajectory when an aircraft has experienced degraded performance due to a failure or damage to the vehicle, the effects of the failure or damage on the lift, drag, and trimmability of the vehicle must be known a priori over the entire flight envelope

This paper investigates the performance of current-mode chaotic neural networks that are optimized for travelling salesman problem (TSP). Piece-wise linear (PWL) functions used as the output functions in chaotic neurons are found to ease the design of neural network circuits. Three different PWL output functions are compared by solving 10-city TSP. Simulation results showed excellent performance of the chaotic neural networks with PWL output function. The design parameters to implement the PWL output function suitable for TSP chaotic neural networks are obtained.

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.

In this paper we apply one of the main results from the theory of noninvertible maps to predict the oscillations amplitude of a chaotic attractor, using only the lines where the piece-wise linear (PWL) map is not differentiable, their first iterates (called critical lines or curves), and their finite-rank iterates. This approach is also valid for smooth differentiable noninvertible maps, where the critical lines are defined as the first iterates of the set of points for which the Jacobian determinant of the map cancels. Moreover, a novel bifurcation encountered in the case of PWL map is presented: chaotic orbit arising from the collision of a typical bifurcation for smooth maps (the Neimark-Hopf, or center bifurcation) with the bifurcation typical only for piece-wise smooth and non differentiable maps (the border-collision bifurcation). Finally, it is shown that this novel bifurcation can give birth to two (or more) chaotic attractors, and the conditions allowing to predict the number of chaotic attractors which could coexist are proposed and discussed in the context of possible applications.

The paper studies a second order nonlinear differential equation whose right hand side is a piecewise linear function. It proves the coexistence of a countably set of periodic solutions and an uncountable set of bounded non-periodic solutions. The Weierstrass approximation theorem is applied so that the piecewise linear function can be approximated by a polynomial on the finite interval. The result can be also used to explain the chaos on some smooth nonlinear dynamical systems.

A set of piecewise-linear (PL) basis functions for signal or functional decomposition is introduced. These basis functions provide a PL approximation to the signal and an a priori determination of the required number for a finite term expansion to achieve a certain pointwise approximation error. Moreover, the expansion coefficients are linear combinations of samples of the function to be expanded and are virtually trivial to determine.

Composite lifting of group inequalities and an application to two-row mixing inequalities

Hybrid systems built up with CMOS devices and Single-Electron Transistors (SET), have been foreseen as a potential alternative in the long term when CMOS downscaling reaches its physical fundamental limits. Although there are still many problems in the fabrication of these nanoelectronic devices, the design flow for hybrid systems requires further development not only in those aspects directly linked to the design and synthesis, but also in aspects regarding the verification of the design, such as electrical simulation. This paper introduces an appropriate model for the SET that can be easily implemented for co-simulation of hybrid systems. The model constitutes a functional model for the SET in the form of an explicit piecewise linear (PWL) formulation that can be easily coded into a high level language.

A novel f-f model is developed constructively which can express any n-dimensional piecewise linear (PWL) function by a superposition of basis functions if it is defined over an nth-order degenerate intersection formed by (n-1)th-order minimal degenerate intersections. We also propose the concrete functional forms of nth-order basis functions. Being the simplest type of the minimal degenerate intersection, the basis function is the most elementary "building block" of a PWL function defined in an arbitrary-dimensional space. In addition, the model constitutes a natural continuation to Julian's canonical formulation and can bridge the lattice PWL model and the well-established canonical representation.

A scheduling problem with piecewise linear (PL) optimization extends conventional scheduling by imposing a conjunction of combinatorial PL constraints involving the objective function variables. To solve this problem, this paper presents a hybrid algorithm where Constraint Programming (CP) search is supported and driven by a (integer) linear programming solver running on a well-controlled subproblem which is dynamically tightened. The paper discusses and compares different ways of decomposing the problem constraints between the CP search and the solver. We show how the subproblem structure and the piecewise linearity are exploited by the search.

In this paper, mapping networks will be considered from the viewpoint of the piecewise-linear (PWL) approximation. The so-called canonical representation plays a kernel role in the PWL representation theory. While this theory has been researched intensively in the contents of mathematics and circuit simulations, little has been seen in the research area about the theoretical aspect of neural networks. This paper modifies this theory and applies it as a mathematical support for mapping networks. The main modification is a "higher-level" generalization of the canonical representation with proofs of its availability in the set of PWL functions. The modified theory will first be used to study the canonical PWL feature of the popular multilayer perceptron-like (MLPL) networks. Second, it will be seen that the generalized canonical representation is itself suitable for a network implementation, which is called the standard canonical PWL network. More generally, the family of (generalized) canonical PWL networks is defined as those which may take the canonical PWL representation as a mathematical model. This family is large and practically meaningful. The standard canonical PWL networks may be taken as representatives in the family. The modification of the PWL representation theory as well as the introduction of this theory in the theoretical study of mapping networks, which provide a new concept of mapping networks, i.e., the canonical PWL network family, may be regarded as the main contributions of the paper.

Hardware implementation of new neuron models or improved conventional neuron models has made a significant contribution to neuromorphic development. One of the important factors considered to improve the conventional neuron models is to explore the impact of electromagnetic energy on neurons. In this work the efficient FPGA implementation of memristive Wilson (MW) neuron model using two approximate MW model is presented. For the first approximate MW (AMW1) model in a hybrid method, piecewise linear (PWL) and CORDIC functions have been used to provide a multiplierless and accurate model. The PWL approximation method is used to provide the second approximate MW (AMW2) model. Results of the FPGA implementation for both the MW and AMW models illustrate that, the AMW1 model with an overall saving of 79%, and the AMW2 model with an overall saving of 69% are appropriate options for large scale implementations. The average NRMSE for the AMW1 model is 0.57%, while for the AMW2 model it is 1.23%. The maximum frequency of AMW2 model is 91.5% better than AMW1 model and realizes high frequency implementation.