Numerical Procedures in Multiobjective Optimization with Variable Ordering Structures

Multiobjective optimization problems with a variable ordering structure, instead of a partial ordering, have recently gained interest due to several applications. In the previous years, a basic theory has been developed for such problems. The binary relations of a variable ordering structure are defined by a cone-valued map that associates, with each element of the linear space ? ^m , a pointed convex cone of dominated or preferred directions. The difficulty in the study of the variable ordering structures arises from the fact that the binary relations are in general not transitive. In this paper, we propose numerical approaches for solving such optimization problems. For continuous problems a method is presented using scalarization functionals, which allows the determination of an approximation of the infinite optimal solution set. For discrete problems the Jahn–Graef–Younes method, known from multiobjective optimization with a partial ordering, is adapted to allow the determination of all optimal elements with a reduced effort compared to a pairwise comparison.     

Optimal Elements in Vector Optimization with a Variable Ordering Structure

Optimality concepts for vector optimization problems with a variable ordering structure are examined. These considerations are motivated by an application in medical image registration, where the preferences vary depending on the element in the image space.     

Crystallographic rationalization of the reactivity and spectroscopic properties of (2R)‐S‐(2,5‐dihydroxyphenyl)cysteine

At 150 K, the title compound, C₉H₁₁NO₄S, crystallizes in the orthorhombic form as a zwitterion and has a low gauche conformation [χ = −46.23 (16)°] for an acyclic cysteine derivative. A difference in bond length is observed for the alkyl C—S bond [1.8299 (15) Å] and the aryl C—S bond [1.7760 (15) Å]. The –NH₃⁺ group is involved in four hydrogen bonds, two of which are intermolecular and two intramolecular. The compound forms an infinite three‐dimensional network constructed from four intermolecular hydrogen bonds. Characterization data (¹³C NMR, IR and optical rotation) are reported to supplement the incomplete data disclosed previously in the literature.     

Frames and Coorbit Theory on Homogeneous Spaces with a Special Guidance on the Sphere

The topic of this article is a generalization of the theory of coorbit spaces and related frame constructions to Banach spaces of functions or distributions over domains and manifolds. As a special case one obtains modulation spaces and Gabor frames on spheres. Group theoretical considerations allow first to introduce generalized wavelet transforms. These are then used to define coorbit spaces on homogeneous spaces, which consist of functions having their generalized wavelet transform in some weighted L_p space. We also describe natural ways of discretizing those wavelet transforms, or equivalently to obtain atomic decompositions and Banach frames for the corresponding coorbit spaces. Based on these facts we treat aspects of nonlinear approximation and show how the new theory can be applied to the Gabor transform on spheres. For the S¹ we exhibit concrete examples of admissible Gabor atoms which are very closely related to uncertainty minimizing states.