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Description:
This article was supported by the German Research Foundation (DFG) and the Open Access Publication Fund of Humboldt-Universität zu Berlin. ; For piecewise linear functions f:Rn↦R we show how their abs-linear representation can be extended to yield simultaneously their decomposition into a convex fˇ and a concave part fˆ , including a pair of generalized gradients gˇ∈Rn∋gˆ . The latter satisfy strict chain rules and can be computed in the reverse mode of algorithmic differentiation, at a small multiple of the cost of evaluating f itself. It is shown how fˇ and fˆ can be expressed as a single maximum and a single minimum of affine functions, respectively. The two subgradients gˇ and −gˆ are then used to drive DCA algorithms, where the (convex) inner problem can be solved in finitely many steps, e.g., by a Simplex variant or the true steepest descent method. Using a reflection technique to update the gradients of the concave part, one can ensure finite convergence to a local minimizer of f, provided the Linear Independence Kink Qualification holds. For piecewise smooth objectives the approach can be used as an inner method for successive piecewise linearization. ; Peer Reviewed