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.

A Robust Adaptive Method for a Quasi-Linear One-Dimensional Convection-Diffusion Problem

Abstract: A quasi-linear conservative convection-diffusion two-point boundary value problem is considered. To solve it numerically, an upwind finite difference scheme is applied. The mesh used has a fixed number (N+1) of nodes and is initially uniform, but its nodes are moved adaptively using a simple algorithm of de Boor based on equidistribution of the arc-length of the current computed piecewise linear solution. It is proved for the first time that a mesh exists that equidistributes the arc-length along the polygonal solution curve and that the corresponding computed solution is first-order accurate, uniformly in $\varepsilon$, where $\varepsilon$ is the diffusion coefficient. In the case when the boundary value problem is linear, if N is sufficiently large independently of $\varepsilon$, it is shown that after $O({\rm ln}(1/\varepsilon)/{\rm ln} N)$ iterations of the algorithm, the piecewise linear interpolant of the computed solution achieves first-order accuracy in the $L^\infty[0,1]$ norm uniformly in $\varepsilon$. Numerical experiments are presented that support our theoretical results.

On Test Functions for Evolutionary Multi-objective Optimization

Abstract: In order to evaluate the relative performance of optimization algorithms benchmark problems are frequently used. In the case of multi-objective optimization (MOO), we will show in this paper that most known benchmark problems belong to a constrained class of functions with piecewise linear Pareto fronts in the parameter space. We present a straightforward way to define benchmark problems with an arbitrary Pareto front both in the fitness and parameter spaces. Furthermore, we introduce a difficulty measure based on the mapping of probability density functions from parameter to fitness space. Finally, we evaluate two MOO algorithms for new benchmark problems.

Modeling disjunctive constraints with a logarithmic number of binary variables and constraints

Abstract: Many combinatorial constraints over continuous variables such as SOS1 and SOS2 constraints can be interpreted as disjunctive constraints that restrict the variables to lie in the union of m specially structured polyhedra. Known mixed integer binary formulations for these constraints have a number of binary variables and extra constraints that is linear in m. We give sufficient conditions for constructing formulations for these constraints with a number of binary variables and extra constraints that is logarithmic in m. Using these conditions we introduce the first mixed integer binary formulations for SOS1 and SOS2 constraints that use a number of binary variables and extra constraints that is logarithmic in the number of continuous variables. We also introduce the first mixed integer binary formulations for piecewise linear functions of one and two variables that use a number of binary variables and extra constraints that is logarithmic in the number of linear pieces of the functions. We prove that the new formulations for piecewise linear functions have favorable tightness properties and present computational results showing that they can significantly outperform other mixed integer binary formulations.