Optimizing Nonsmooth Lipschitz Continuous Functions by Piecewise Linearization Based on Algorithmic Differentiation

Sabrina Fiege*, Andreas Griewank and Andrea Walther

From various application as e.g. engineering, economies and other fields nonsmooth objective functions arise. In this talk a novel optimization approach based on Algorithmic Differentiation (AD) is presented that exploits the structure of the function. We consider locally Lipschitz continuous and piecewise differentiable functions. Therefore, the standard AD approach was extended by abs(), min(), and max(). To optimize the nonsmooth functions we approximate them locally by a piecewise-linear model. This model will have a finite number of kinks between open polyhedral facets decomposing the function domain. On each facet we repeatedly solve quadratic subproblems which are compositions of the linearized target function and a quadratic overestimation. To switch between the different facets we exploit the structure given by the decomposition of the domain. This causes disjunctive quadratic subproblems. This talk presents the described optimization method in detail as well as first numerical results.

Mathematics Subject Classification: 90C30

Keywords: Algorithmic Differentiation; Nonsmooth Optimization

Minisymposion: Nonsmooth Optimization