Solving piecewise linear systems in abs-normal form

Abstract

With the ultimate goal of iteratively solving piecewise smooth (PS) systems, we consider the solution of piecewise linear (PL) equations. As shown in [Gri13] PL models can be derived in the fashion of automatic or algorithmic differentiation as local approximations of PS functions with a second order error in the distance to a given reference point. The resulting PL functions are obtained quite naturally in what we call the abs-normal form, a variant of the state representation proposed by Bokhoven in his dissertation [vB81]. Apart from the tradition of PL modelling by electrical engineers, which dates back to the Master thesis of Thomas Stern [Ste56] in 1956, we take into account more recent results on linear complementarity problems and semi-smooth equations originating in the optimization community [CPS92, Sch12, FP03]. We analyze simultaneously the original PL problem (OPL) in abs-normal form and a corresponding complementary system (CPL), which is closely related to the absolute value equation (AVE) studied by Mangasarian et al [MM06] and a corresponding linear complementarity problem (LCP). We show that the CPL, like KKT conditions and other simply switched systems, cannot be open without being injective. Hence some of the intriguing PL structure described by Scholtes in [Sch12] is lost in the transformation from OPL to CPL. To both problems one may apply Newton variants with appropriate generalized Jacobians directly computable from the abs-normal representation. Alternatively, the CPL can be solved by Bokhoven's modulus method and related fixed point iterations. We compile the properties of the various schemes and highlight the connection to the properties of the Schur complement matrix, in particular its signed real spectral radius as analyzed by Rump in [Rum97]. Numerical experiments and suitable combinations of the fixed point solvers and stabilized generalized Newton variants remain to be realized.

On Lipschitz optimization based on gray-box piecewise linearization

“…When p = 2 we have a least squares problem, where the polyhedral structure is inherited from F(x) but the quadratic term may jump at the interfaces. The formally well-determined case m = n of piecewise linear equation solving in abs-normal form has recently been studied in [12]. Finding a stationary point of a generalized gradient is the symmetric variant of solving an algebraic inclusion 0 ∈ F(x), where F : R n ⇒ R n is a convex outer semi-continuous multifunction.…”

“…Furthermore, since we wish to minimize, let us assume for simplicity that f (0) = 0. As shown in [12,Sect. 2], any such PL scalar function y = f (x) can be expressed in terms of a so-called switching vector z ∈ R s in the abs-normal form…”

“…As detailed in [11] and [12] in our gray-box scenario, the following information can be readily computed at any pair x, d ∈ R n with d = 0 with an appropriate extended algorithmic differentiation procedure:…”

On Automatic Differentiation and Algorithmic Linearization

“…That situation might take a little getting used to, since one tends to expect that the end product of any differentiation process is some collection of derivative vectors or matrices. In fact, one can describe F(x ; ·) in terms of matrices and vectors by the so-called abs-normal form described in [18]. It can be generated directly by a minor extension of standard AD tools, which has for example been provided for ADOL-C.…”

(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 |.…”