This special issue of Optimization is related to the 6th German Polish Conference on Optimization – Methods and Applications with special emphasis on Smooth and Nonsmooth Optimization, Variational Problems, Variational Inequalities and Optimal Control which took place at the Leucorea (which is part of the Martin-Luther-Universität Halle – Wittenberg) from February 28 to March 4, 2014. It was the 6th conference in the German Polish Conference on Optimization series, which has been held in Zagan (1999), Cottbus (2002), Bedlewo (2005), Moritzburg (2009) and Krakow (2011). Like the previous conferences, the presentations at the conference in Wittenberg covered a broad range of topics in mathematical optimization, as vector and set-valued optimization, optimal control, parametric optimization and sensitivity, non-smooth analysis, as well as applications. Over 50 participants attended 6 plenary and 35 contributed talks. I thank all who made the conference a success, especially Christiane Tammer (Halle), Andrzej Cegielski (Zielona Góra), Andreas Löhne (Jena), Maria Pilecka and David Voigt from Freiberg.
The contribution by G. Eichfelder and T. Gerlach investigates vector optimization problems with variable ordering structures. These problems have recently found many applications e.g. in medicine, economy theory or in behavioural sciences. They introduce and compare various definitions of properly optimal elements for variable ordering structures and suggest a new scalarization approach which can be used for a complete characterization of optimal elements of those problems without convexity assumption.
Properties of non-smooth functions are the topic of the contribution of V.V. Gorokhovik and M.A. Trafimovich. They show first that any continuous positively homogeneous function p can be represented as the pointwise supremum of an one-parametric monotone family of Lipschitz continuous positively homogeneous functions. This is related to the well-known Moreau envelopes and proximal behavior of lower semi-continuous functions. Using this result, they investigate lower (upper) primal (and dual) exhausters which are important tools for a local approximation of non-smooth functions used in non-smooth analysis and optimization.
Topic of the article by S. Pickenhain and co-authors are special optimal control problems with infinite horizon for which less results, especially solution algorithms, are known as for problems with finite horizon. The authors prove a Pontryagin-type maximum principle which is not only a necessary optimality condition but also a sufficient one for the considered class of problems. Based on this principle, a pseudo-spectral solution algorithm is suggested, numerical results are discussed.
The equation of is of crucial interest not only in game theory and with respect to duality in optimization. Assuming generalized convexity, M. Syga proves this equation in her article using a weak
-intersection property. Necessary conditions guaranteeing that this property is satisfied are formulated, examples are discussed. No compactness of the sets X, Y is needed.
Convex sets can equivalently be described using their (Lipschitz continuous) support function. A thorough investigation of the relations between the differentiability of the support function on the interior of its domain and non-emptyness of the interior/convexity of the set can be found in the article by C. Zălinescu. The results are then applied to differentiability of the cost function in economics.
The papers in this issue underwent a regular review process according to the rules for the journal Optimization. I am convinced that they all provide interesting scientific contributions (to different research fields). I am thankful to the reviewers for their time and suggestions and to the authors for their contributions.
On behalf of the organizers and the participants of the conference, I want to express our gratitude to the Deutsche Forschungsgemeinschaft and the Deutsche Bank as well as to both Martin-Luther-Universität Halle-Wittenberg and Technische Universität Bergakademie Freiberg for support.
On behalf of the steering committee
TU Bergakademie Freiberg, Germany
[email protected]
© 2016, Stephan Dempe