Publikationen

We present a branch-and-bound algorithm for minimizing multiple convex quadratic objective functions over integer variables. Our method looks for efficient points by fixing subsets of variables to integer values and by using lower bounds in the form of hyperplanes in the image space derived from the continuous relaxations of the restricted objective functions. We show that the algorithm stops after finitely many fixings of variables with detecting both the full efficient and the nondominated set of multiobjective strictly convex quadratic integer problems. A major advantage of the approach is that the expensive calculations are done in a preprocessing phase so that the nodes in the branch-and-bound tree can be enumerated fast. We show numerical experiments on biobjective instances and on instances with three and four objectives.

Similarity measurement of two probability distributions is important in many applications of statistics. Embedding such distributions into a reproducing kernel Hilbert space (RKHS) has many favorable properties. The choice of the reproducing kernel is crucial in the approach. We study this question by considering the similarity of two distributions of the same class. In particular, we investigate when the RKHS embedding is "admissible" in the sense that the distance between the embeddings should become smaller when the expectations are getting closer or when the variance is increasing to infinity. We give conditions on the widely-used translation-invariant reproducing kernels to be admissible. We also extend the study to multivariate non-symmetric Gaussian distributions.

Current generalizations of the central ideas of single-objective branch-and-bound to the multiobjective setting do not seem to follow their train of thought all the way. The present paper complements the various suggestions for generalizations of partial lower bounds and of overall upper bounds by general constructions for overall lower bounds from partial lower bounds, and by the corresponding termination criteria and node selection steps. In particular, our branch-and-bound concept employs a new enclosure of the set of nondominated points by a union of boxes. On this occasion we also suggest a new discarding test based on a linearization technique. We provide a convergence proof for our general branch-and-bound framework and illustrate the results with numerical examples.