Direct solution of piecewise linear systems

Abstract

Let S be a real n × n matrix, z,ĉ ∈ R n , and |z| the componentwise modulus of z. Then the piecewise linear equation system z − S|z| =ĉ is called an absolute value equation (AVE). It has been proven to be equivalent to the general linear complementarity problem, which means that it is NP hard in general.We will show that for several system classes the AVE essentially retains the good natured solvability properties of regular linear systems. I.e., it can be solved directly by a slightly modified Gaussian elimination that we call the signed Gaussian elimination. For dense matrices S this algorithm has the same operations count as the classical Gaussian elimination with symmetric pivoting. For tridiagonal systems in n variables its computational cost is roughly that of sorting n floating point numbers. The sharpness of the proposed restrictions on S will be established.

Convergence results for some piecewise linear solvers

“…Proof. (1) ⇒ (4) : ρ R (A) < 1 implies that the real eigenvalues of all (I − AS), S ∈ S n , are positive and no complex eigenvalue is 0 [Rad16a].…”

“…The latest publications on that matter include approaches by linear programming [Man14] and concave minimization [Man07a], as well as a variety of Newton and fixed point methods (see, e.g., [BC08], [YY12], [HHZ11]). In this article we will present and further analyze two solvers for the AVE: the signed Gaussian elimination, which is a direct solver that was developed in [Rad16a]; and a semi-iterative generalized Newton method developed in [GBRS15,SGRB14]. In particular, we unify and further extend the known convergence results for both algorithms.…”

An Open Newton Method for Piecewise Smooth Functions

“…In the situation of Theorem 4.4 the Newton steps can be calculated in (weakly) polynomial time via interior point methods for LCPs [Pot07], using the equivalence of AVE and LCP. Algorithms which are more efficient for special system structures can be found, e.g., in [BC09], [GBRS15], [Rad16].…”

(Almost) matrix‐free solver for piecewise linear functions in abs‐normal form

“…Based on the specific structure of the compact representation, several methods were developed to find a solution of an ANF if n = m and given , that is, compute such that Δ y =Δ F ( x ,Δ x ) or, respectively, holds. Two simple methods are the simple modulus‐like iteration and the signed fixed‐point iteration to find a solution of the fixed‐point equation Δ z =[ a − ZJ −1 ( b −Δ y )]+ S |Δ z |—a rigourous derivation of the stated fixed‐point equation/iterations, their connection to closely related linear complementarity problems (LCPs), and the absolute value equation besides other methods can be found in the cited references. Here, represents the Schur complement of the ANF's system matrix, and is the diagonal matrix containing the signatures of the current switching variable Δ z on its main diagonal such that ∑ Δ z Δ z =|Δ z |.…”

“…to find a solution of the fixed-point equation Δz = [a − ZJ −1 (b − Δy)] + S|Δz|-a rigourous derivation of the stated fixed-point equation/iterations, their connection to closely related linear complementarity problems (LCPs) 8 , and the absolute value equation 9 besides other methods [10][11][12][13][14] can be found in the cited references. Here,…”