Solving piecewise linear equations in abs-normal form

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 modeling 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 unfolded system (UPL), which is closely related to a linear complementarity problem (LCP). We show that the UPL, like KKT conditions and other singly switched systems, cannot be open without being injective. Hence we find that some of the intriguing PL structure described by Scholtes [Sch12] is lost in the unfolding from OPL to UPL. To both problems one may apply Newton variants with appropriate generalized Jacobians directly computable from the abs-normal representation. Alternatively, the ULP can be solved by Bokhoven’s modulus method and another fixed point solver that is asymptotically faster but requires a slightly stronger convergence condition. 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 done. Preprint submitted to Linear Algebra and its Applications November 21, 2013