Publications

There are many generalizations of Ekeland's variational principle for vector optimization problems with fixed ordering structures, i.e., ordering cones. These variational principles are useful for deriving optimality conditions, [epsilon]-Kolmogorov conditions in approximation theory, and [epsilon]-maximum principles in optimal control. Here, we present several generalizations of Ekeland's variational principle for vector optimization problems with respect to variable ordering structures. For deriving these variational principles we use nonlinear scalarization techniques. Furthermore, we derive necessary conditions for approximate solutions of vector optimization problems with respect to variable ordering structures using these variational principles and the subdifferential calculus by Mordukhovich.

We prove new enclosures for the spectrum of non-selfadjoint operator matrices associated with second order linear differential equations z'' (t) + D z' (t) + A_0 z(t) = 0 in a Hilbert space. Our main tool is the quadratic numerical range for which we establish the spectral inclusion property under weak assumptions on the operators involved; in particular, the damping operator only needs to be accretive and may have the same strength as A_0. By means of the quadratic numerical range, we establish tight spectral estimates in terms of the unbounded operator coefficients A_0 and D which improve earlier results for sectorial and selfadjoint D; in contrast to numerical range bounds, our enclosures may even provide bounded imaginary part of the spectrum or a spectral free vertical strip. An application to small transverse oscillations of a horizontal pipe carrying a steady-state flow of an ideal incompressible fluid illustrates that our new bounds are explicit.

In this paper we consider k-server problems with parallel requests where several servers can also be located on one point. We will distinguish the surplussituation where the request can be completely fulfilled by means of the k servers and and the scarcity-situation where the request cannot be completely met. First, we will give an example. It shows that the corresponding work function algorithm is not competitive in the case of the scarcity-situation. Until now, it remains an open question whether the work function algorithm is competitive or not in the case of the surplus-situation. Thats why, we will suggest the new "compound work function algorithm" in the following section and prove that this algorithm is also (2 k - 1)-competitive.