BibTeX
@ARTICLE{
         Griewank2015Spl,
       doi = "10.1016/j.laa.2014.12.017",
       keywords = "Switching depth, Sign real spectral radius, Coherent orientation, Generalized
         Jacobian, Semismooth Newton, Partial contractivity, Complementary system",
       abstract = "With the ultimate goal of iteratively solving piecewise smooth (PS) systems, we
         consider the solution of piecewise linear (PL) equations. As shown in [7] 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 [27]. Apart from the tradition of PL modelling by
         electrical engineers, which dates back to the Master thesis of Thomas Stern [26] in 1956, we take
         into account more recent results on linear complementarity problems and semi-smooth equations
         originating in the optimization community [3], [25], [5]. 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 and Meyer [14] 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 [25] 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 partial
         contractivity as analyzed by Rohn and Rump [23]. Numerical experiments and suitable combinations of
         the fixed point solvers and stabilized generalized Newton variants remain to be realized.",
       author = "Griewank, Andreas and Bernt, Jens-Uwe and Radons, Manuel and Streubel, Tom",
       year = "2015",
       month = "04",
       pages = "500--530",
       title = "Solving piecewise linear systems in abs-normal form",
       volume = "471",
       journal = "Linear Algebra and its Applications",
       ad_theotech = "Piecewise Linear"
}