Polyhedral Dc Decomposition And Dca Optimization Of Piecewise Linear Functions

For piecewise linear functionsf:R(n)bar right arrow R we show how their abs-linear representation can be extended to yield simultaneously their decomposition into a convex f and a concave part (f) over cap, including a pair of generalized gradients g is an element of R-n (g) over cap. 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 f and (f) over cap can be expressed as a single maximum and a single minimum of affine functions, respectively. The two subgradients g and -(g) over cap 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.