Direct solution of piecewise linear systems

Iterative Solution of Piecewise Linear Systems

Abstract. The correct formulation of numerical models for free-surface hydrodynamics often requires the solution of special linear systems whose coefficient matrix is a piecewise constant function of the solution itself. In so doing one may prevent the development of unrealistic negative water depths. The resulting piecewise linear systems are equivalent to particular linear complementarity pro...

Piecewise Constant Controlled Linear Fuzzy Systems

Piecewise Linear Solution Paths for Para- metric Piecewise Quadratic Programs with Application to Direct Weight Optimization

Recently, pathfollowing algorithms for parametric optimization problems with piecewise linear solution paths have been developed within the field of regularized regression. This paper presents a generalization of these algorithms to a wider class of problems, namely a class of parametric piecewise quadratic programs and related problems. It is shown that the approach can be applied to the nonpa...

Piecewise Linear Regularized Solution Paths

We consider the generic regularized optimization problem β̂(λ) = arg minβ L(y,Xβ) + λJ (β). Efron, Hastie, Johnstone and Tibshirani [Ann. Statist. 32 (2004) 407–499] have shown that for the LASSO—that is, if L is squared error loss and J (β)= ‖β‖1 is the 1 norm of β—the optimal coefficient path is piecewise linear, that is, ∂β̂(λ)/∂λ is piecewise constant. We derive a general characterization of ...

Minimal solution of fuzzy linear systems

In this paper, we use parametric form of fuzzy number and we converta fuzzy linear system to two linear system in crisp case. Conditions for the existence of a minimal solution to $mtimes n$ fuzzy linear equation systems are derived and a numerical procedure for calculating the minimal solution is designed. Numerical examples are presented to illustrate the proposed method.