On Optimality Conditions for Constrained Nonsmooth Optimization Based on the Directed Subdifferential

Wolfgang Achtziger, R. Baier, E. Farkhi, V. Roshchina

We consider the directed subdifferential based on the concept of directed sets as suggested by Baier and Farkhi in 2008. The beauty of directed sets lies in the formal straightforwardness
of the arithmetic calculus for embedded convex compact sets. As a consequence, direct and precise calculus rules apply for directed subdifferentials of sums, differences, and the pointwise maximum of nonsmooth functions. In the recent past necessary and sufficient optimality conditions based on the directed subdifferential have been suggested for unconstrained optimization of, e.g., quasidifferentiable functions. In this talk we present some results on standard approaches tackling constrained problems through optimality conditions of unconstrained optimization. We investigate the exact l1-penalty approach, Lagrange duality, and saddle point optimality conditions. The
theoretical results are illustrated at hand of academic examples.