Prof. Andreas Griewank, School of Mathematics and Computer Science, Yachay Tech

Title:
Optimality and convexity conditions for piecewise smooth objective functions

Abstract:

Any piecewise smooth function that is specified by an evaluation procedure involving smooth elemental functions and piecewise linear functions like min and max can be represented in abs normal form. This is in particular true for most popular nonsmooth test functions. By an extension of algorithmic, or automatic differentiation, one can then compute certain first and second order derivative vectors and matrices that represent a local piecewise linearization and provide additional curvature information. On the basis of these quantities we characterize local optimality by first and second order necessary and sufficient conditions, which generalize the corresponding KKT and SSC theory for smooth problems. The key assumption is the Linear Independence Kink Qualification (LIKQ), a generalization of LICQ familiar from NLOP. It implies that the objective has locally a so-called VU decomposition and renders everything tractable in terms of matrix factorizations and other simple linear algebra operations.