Daniel Scholz

The big cube small cube solution method for multidimensional facility location problems

In this paper we propose a general solution method for (non-differentiable) facility location problems with more than two variables as an extension of the Big Square Small Square technique (BSSS). We develop a general framework based on lower bounds and discarding tests for every location problem. We demonstrate our approach on three problems: the Fermat-Weber problem with positive and negative weights, the median circle problem, and the p-median problem. For each of these problems we show how to calculate lower bounds and discarding tests. Computational experiences are given which show that the proposed solution method is fast and exact.

The theoretical and empirical rate of convergence for geometric branch-and-bound methods

Geometric branch-and-bound solution methods, in particular the big square small square technique and its many generalizations, are popular solution approaches for non-convex global optimization problems. Most of these approaches differ in the lower bounds they use which have been compared empirically in a few studies. The aim of this paper is to introduce a general convergence theory which allows theoretical results about the different bounds used. To this end we introduce the concept of a bounding operation and propose a new definition of the rate of convergence for geometric branch-and-bound methods. We discuss the rate of convergence for some well-known bounding operations as well as for a new general bounding operation with an arbitrary rate of convergence. This comparison is done from a theoretical point of view. The results we present are justified by some numerical experiments using the Weber problem on the plane with some negative weights.

A note on the Zassenhaus product formula

We provide a simple method for the calculation of the terms cn in the Zassenhaus product ea+b=ea∙eb∙∏n=2∞ecn for noncommuting a and b. This method has been implemented in a computer program. Furthermore, we formulate a conjecture on how to translate these results into nested commutators. This conjecture was checked up to order n=17 using a computer.

Computing the Baker–Campbell–Hausdorff series and the Zassenhaus product

The Baker–Campbell–Hausdorff (BCH) series and the Zassenhaus product are of fundamental importance for the theory of Lie groups and their applications in physics and physical chemistry. Standard methods for the explicit construction of the BCH and Zassenhaus terms yield polynomial representations, which must be translated into the usually required commutator representation. We prove that a new translation proposed recently yields a correct representation of the BCH and Zassenhaus terms. This representation entails fewer terms than the well-known Dynkin–Specht–Wever representation, which is of relevance for practical applications. Furthermore, various methods for the computation of the BCH and Zassenhaus terms are compared, and a new efficient approach for the calculation of the Zassenhaus terms is proposed. Mathematica implementations for the most efficient algorithms are provided together with comparisons of efficiency.

Theoretical rate of convergence for interval inclusion functions

Geometric branch-and-bound methods are commonly used solution algorithms for non-convex global optimization problems in small dimensions, say for problems with up to six or ten variables, and the efficiency of these methods depends on some required lower bounds. For example, in interval branch-and-bound methods various well-known lower bounds are derived from interval inclusion functions. The aim of this work is to analyze the quality of interval inclusion functions from the theoretical point of view making use of a recently introduced and general definition of the rate of convergence in geometric branch-and-bound methods. In particular, we compare the natural interval extension, the centered form, and Baumann’s inclusion function. Furthermore, our theoretical findings are justified by detailed numerical studies using the Weber problem on the plane with some negative weights as well as some standard global optimization benchmark problems.

A solution algorithm for non-convex mixed integer optimization problems with only few continuous variables

Geometric branch-and-bound techniques are well-known solution algorithms for non-convex continuous global optimization problems with box constraints. Several approaches can be found in the literature differing mainly in the bounds used.

Geometric branch-and-bound methods for constrained global optimization problems

Geometric branch-and-bound methods are popular solution algorithms in deterministic global optimization to solve problems in small dimensions. The aim of this paper is to formulate a geometric branch-and-bound method for constrained global optimization problems which allows the use of arbitrary bounding operations. In particular, our main goal is to prove the convergence of the suggested method using the concept of the rate of convergence in geometric branch-and-bound methods as introduced in some recent publications. Furthermore, some efficient further discarding tests using necessary conditions for optimality are derived and illustrated numerically on an obnoxious facility location problem.

Approximately disentangling exponential operators

A new method for the approximate disentangling of exponential operators based on the Baker–Campbell–Haussdorff theorem is suggested and implemented in a computer program. The operators to be disentangled must form a finite-dimensional Lie algebra. The accuracy of the method is tested and demonstrated in several explicitly calculated examples, where exact analytic solutions are available.

Principles and basic concepts

In this chapter, our main goal is to summarize principles and basic concepts that are of fundamental importance in the remainder of this text, especially in Chapter 3 where bounding operations are presented.We begin with the definition of convex functions and some generalizations of convexity in Section 1.1. Some fundamental but important results are given before we discuss subgradients. Next, in Section 1.2 we briefly introduce distance measures given by norms. Distance measures are quite important in the subsequent Section 1.3 where we give a very brief introduction to location theory. Furthermore, we show how to solve theWeber problem for the rectilinear and Euclidean norms. Moreover, d.c. functions are introduced in Section 1.4 and basic properties are collected. Finally, we give an introduction to interval analysis in Section 1.5 which leads to several bounding operations later on.

Studying Photon Antibunching of Bunched Emitters

We report on the single photon emission of bunched NV-centres by focusing on different spatial fractions of the emission spot, which shows that g(2)(0)<0.5 does not sufficiently prove the single photon characteristics of the centres.