Foreword

The Seventh German-Polish Conference on Optimization, GPCO 2017, took place on August 27–September 1, 2017, at the Mathematical Research and Conference Center of the Institute of Mathematics of the Polish Academy of Sciences, Będlewo, near Poznań. Previous German-Polish conferences on optimization were held in Żagań (1999), Cottbus (2002), Będlewo (2005), Moritzburg (2009), Kraków (2011) and Wittenberg (2014).

The 7th GPCO 2017 conference focused on, but was not limited to, the following topics:

• smooth and nonsmooth optimization,

• variational analysis,

• inverse and ill posed problems,

• solution methods.

Over 30 contributions were presented on these and related topics. The following selection of contributions are available in the present proceedings volume.

1. Francesca Bucci and Irena Lasiecka. The Jordan–Moore–Gibson–Thompson equation describes the acoustic velocity potential in ultrasound wave propagation. A great deal of attention has been recently devoted to its linearization referred to in the literature as the Moore–Gibson–Thompson equation. In this work, we consider and solve a quadratic control problem associated with the linear equation, formulated consistently with the goal of keeping the acoustic pressure close to a reference pressure during ultrasound excitation, as required in medical and industrial applications.

2. Radu Ioan Bot and Dang-Khoa Nguyen. We investigate a forward-backward splitting algorithm of penalty type with inertial effects for finding the zeros of the sum of a maximally monotone operator and a cocoercive one and the convex normal cone to the set of zeroes of an another cocoercive operator. Weak ergodic convergence is obtained for the iterates, provided that a condition expressed via the Fitzpatrick function of the operator describing the underlying set of the normal cone is verified. Under strong monotonicity assumptions, strong convergence for the sequence of generated iterates is proved.

3. Zijia Peng, Leszek Gasiński, Stanisław Migórski and Anna Ochal. We study a nonconvex constrained evolution problem for a set being the union of a finite number of convex sets. The problem generalizes the classical parabolic variational inequality of the second kind and contains an operator of L-(S)+-type. The existence result for this problem is proved by using a variational-hemivariational inequality approach, a surjectivity theorem for multi-valued pseudomonotone operators in a reflexive Banach space and a penalization method in which a small parameter does not have to tend to zero.

4. Akhtar A. Khan, Stanisław Migórski and Miguel Sama. We study the inverse problem of identifying a variable parameter in variational and quasi-variational inequalities. We consider a quasi-variational inequality involving a multi-valued monotone map and give a new existence result. We then formulate the inverse problem as an optimization problem and prove its solvability. We also conduct a thorough study of the inverse problem of parameter identification in noncoercive variational inequalities which appear commonly in applied models.

5. Valentin V. Gorokhovik. The primary goal of the paper is to establish characteristic properties of (extended) real-valued functions defined on normed vector spaces that admit the representation as the lower envelope (the pointwise infimum) of their minimal (with respect of the pointwise ordering) convex majorants. The results presented generalize and extend the well-known Demyanov-Rubinov characterization of upper semicontinuous positively homogeneous functions (as the lower envelope of exhaustive families of continuous sublinear functions) to larger classes of functions (not necessarily positively homogeneous).

6. Muhammad Abid Dar, Andreas Fischer, John Martinovic and Guntram Scheithauer. The minimum connectivity inference (MCI) problem represents an NP-hard generalization of the well-known minimum spanning tree problem and has been studied in different fields of research independently. Whereas, in general, existing approaches can only be applied to find approximate solutions or optimal edge sets of rather small instances, concepts to optimally cope with more meaningful problem sizes have not been proposed yet in literature. For this reason, we present a new mixed integer linear programming formulation for the MCI problem, and introduce new instance reduction methods that can be applied to downsize the complexity of a given instance prior to the optimization.

7. Abebe Geletu, Armin Hoffmann and Pu Li. An inner-outer approximation approach was recently developed to solve single chance constrained optimization (SCCOPT) problems. We extend this approach to address joint chance constrained optimization (JCCOPT) problems. Using an inner-outer approximation, two smooth parametric optimization problems are defined whose feasible sets converge to the feasible set of JCCOPT from inside and outside, respectively. The continuous differentiability of the probability function of a joint chance constraint is obtained by examining the uniform convergence of the gradients of the parametric approximations.

8. Ewa Bednarczuk and Krzysztof W. Leśniewski. We prove that a strongly cone paraconvex mapping defined on a normed space X and taking values in a reflexive separable Banach space Y is Gâteaux differentiable on a dense G-delta subset of X. We also discuss Fréchet differentiability in the case when X is an Asplund space. Our results are generalizations of Rolewicz theorem (Theorem 3.1 from the paper by Rolewicz, Differentiability of strongly paraconvex vector-valued functions. Funct Approx. 2011;2:273-277).

9. Daniel Ciripoi, Andreas L$\ddot{o}$hne and Benjamin Weissing. The article deals with operations defined on convex polyhedra or polyhedral convex functions. Given two convex polyhedra, operations like Minkowski sum, intersection and closed convex hull of the union are considered. Basic operations for one convex polyhedron are, for example, the polar, the conical hull and the image under affine transformation. The concept of a P-representation of a convex polyhedron is introduced. It is shown that many polyhedral calculus operations can be expressed explicitly in terms of P-representations.

10. Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański. Having a function being a difference of sublinear functions defined on a plane, we present a formula for effective calculation of sublinear functions such that their difference is equal to the given one. Moreover, these newly calculated sublinear functions are minimal and as such unique-up-to-linear-summand. We also provide examples of such functions.

The papers in this proceedings volume underwent a regular review process according to the rules of the journal Optimization. They provide interesting contributions to different research fields and show the diversity of the scientific results presented at the conference. We are thankful to the reviewers for their time and suggestions and to the authors for their contributions.

The conference was attended by 54 participants from Germany, Poland, Austria, Belarus, Czech Republic, France, Italy, Israel, Russia, USA, Vietnam and Iran (https://www.impan.pl/en/activities/banach-center/conferences/17-gpco). It could not be realized without the generous support of the sponsors who contributed to making it all happen: the Warsaw Center of Mathematics and Computer Science, Systems Research Institute of the Polish Academy of Sciences, Banach Center, Institute of Mathematics of the Polish Academy of Sciences.