Nonsmooth optimization by successive abs-linearization in function spaces

Abstract

We present and analyze the solution of nonsmooth optimization problems by a quadratic overestimation method in a function space setting. Under certain assumptions on a suitable local model, we show convergence to first-order minimal points. Subsequently, we discuss an approach to generate such a local model using the so-called abs-linearization. Finally, we discuss a class of PDE-constrained optimization problems incorporating the $L^1$-penalty term that fits into the considered class of nonsmooth optimization problems.

COMMUNICATED BY:

• Jen-Chih Yao

Keywords:

• Abs-linearization
• quadratic overestimation method
• first-order minimality

2010 Mathematics Subject Classifications:

• 49J52
• 65K10
• 47N10
• 35Q93

Acknowledgements

We would like to thank Johannes Lankeit and Gerd Wachsmuth for the helpful comments and useful suggestions that greatly improved the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Furthermore, we acknowledge funding by the DFG priority program SPP 1962 within the project “Shape Optimization for Maxwell's Equations Including Hysteresis Effects in the Material Laws (HyLa)” (WA1607/12-1).