AG Optimierung

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Im Oberseminar halten Mitglieder und Gäste der Arbeitsgruppe Vorträge zu wechselnden Themen der mathematischen Optimierung. Gäste sind jederzeit willkommen. Das Oberseminar findet während der Vorlesungszeit etwa ein- bis zweimal pro Monat

um 17:15 Uhr
in 48-208


Oberseminar im Sommersemester 2023

Im Sommersemester 2023 finden folgende Vorträge statt:

Among a large number of mathematical models that have been developed to address the evacuation problems, the network flows are the central and widely used operational research models in existence. These models are not only important in case of emergency periods, but also essential to address the traffic congestion in everyday rush office hours. An effective solution of them would be very helpful to explore their applicability in practice supporting the SDG goals in the disaster prone countries. This talk will be focused on the overview and insights of the results with objectives of time minimization and flow maximization. Examples will be illustrated to clarify the obtained results.

By the seminal work of Bertsimas & Sim in 2003, we can solve robust combinatorial optimization problems under budget uncertainty in theory by either solving n+1 deterministic problems or a compact LP reformulation introducing only additional continuous variables. Hence, the computational complexity does not increase. In practice however, computation times can increase significantly in comparison to the deterministic instance. In this talk, two approaches to speed-up the solving of such problems are presented. On the one hand, we derive a tailored branch-and-bound algorithm. On the other hand, we show how valid inequalities for the deterministic problem can be recycled to obtain new valid inequalities for the robust problem. This is joint work with Timo Gersing and Christina Büsing.

Der Vortrag findet zur üblichen Zeit jedoch in Raum 48/210 im Rahmen des Felix-Klein-Kolloquiums des Fachbereichs statt.

We regard a natural extension of Hamiltonian graphs: removing a Hamiltonian cycle from a cubic graph leaves a perfect matching. Conversely, removing a perfect matching \(M\) from a cubic graph \(G\) leaves a disjoint union of cycles. Contracting these cycles yields a new graph \(G_M\). The graph G is star-like if \(G_M\) is a star for some perfect matching \(M\), making Hamiltonian graphs star-like. We extend the technique used to prove that Hamiltonian graphs satisfy the 3-decomposition conjecture to show that 3-connected star-like graphs satisfy it as well. Finally, we quickly look at the additional challenges that arise when we attempt to generalise the result to tree-like graphs.

Oberseminar im Wintersemester 2022/23

Im Wintersemester 2022/23 finden folgende Vorträge statt:

In this talk we shall begin by discussing the paper manufacturing process. Followed by this we shall have an overview of various optimization problems that arise from this paper industries. Then we shall restrict ourselves to a specific optimization problem called deckle and trim loss optimization, which is well known as cutting stock problem.

About the speaker:
Dr. A Chandrashekaran did his PhD from Indian Institute of Technology Madras, India in the year 2010. He has two years of Department of Science and Technology, India post doctoral fellowship at the C R Rao Insitute, Hyderabad, India. He is presently an Assistant Professor of Mathematics at the Central University of Tamil Nadu, Thiruvarur, India. His research interests include Optimization, Mathematical Programming and Game Theory.

Der Vortrag findet abweichend bereits um 10:00 Uhr und in Raum 14/420 statt.

In multi-objective optimization (MOO), the goal is to optimize several objectives for a problem at the same time. As a consequence, not one single solution is simply "optimal" anymore. We look for the set of Pareto-optimal solutions instead. One specific subproblem of MOO is Tri-Objective Linear Programming (TOLP). In 2022, the first algorithm with polynomial delay for extreme points of TOLP was presented by Fritz Bökler. This presentation is about a master's thesis in which this algorithm is implemented and developed further. TriPoD is the name of the resulting software and stands for Tri-Objective Polynomial Delay. As the thesis is still ongoing, many practical results are intermediary. The first part of the presentation gives an introduction into MOO and Bökler's new algorithm. Then, we look at the state of the art for practical TOLP solving and compare the performance of the new algorithm to established solvers. This leads us to a refinement of the algorithm: We trade polynomial delay for polynomial space. The polynomial space algorithm introduces the opportunity to parallelize the algorithm. As no current state-of-the-art solver is able to efficiently use parallel computing, this might open completely new possibilities in the future.

We introduce Min Cost \(q\)-Multitype Multiset Multicover. Informally speaking, the goal of the latter is to find a packing of locations such that the region demands can be covered in any scenario without over-stressing the doctor capacities. For the aforementioned problem we develop a dynamic programming algorithm step by step. Moreover, we relate it to the One Plan project for which we also present some new heuristics.

We study multi-modal route planning in a network and present an overview of state-of-the-art algorithms. We sketch the historical development of these algorithms (including preprocessing techniques), explain the algorithmic ideas, and report about running times of implementations of these algorithms. Finally, we discuss current research topics in this area.

We consider the fundamental scheduling problem of minimizing the sum of weighted completion times on a single machine in the non-clairvoyant setting, i.e., with unknown processing times. In this setting a common assumption is that jobs can be arbitrarily preempted and resumed later without any overhead. I will start my talk with a proof of the 2-competitiveness of the Weighted Round-Robin Algorithm, a classical result of Kim and Chwa (2003). Then I will give an overview of our new results. First, the 2-competitiveness can be generalized to the setting where jobs arrive online over time. Second, we study the alternative model in which jobs cannot be suspended but only killed and restarted later. For this setting, we performed a tight analysis of the natural scaling strategy and a randomized variant, resulting in competitive ratios of ≈6.197 and ≈3.032, respectively. This strategy can be extended to jobs arriving online as well as scheduling unit-weight jobs on identical parallel machines, for which upper bounds < 10 on the competitive ratio are obtained. This is joint work with Guillaume Sagnol, Daniel Schmidt genannt Waldschmidt, and Philipp Warode from TU Berlin.

See also:

A major bottleneck for exact solution methods of multiobjective optimization problems is that the cardinalities of the set of nondominated images and the set of efficient solutions may be exponentially large or even infinite. This strongly motivates the approximation of multiobjective optimization problems - a concept to substantially reduce the number of required solutions while still obtaining a provable solution quality. Here, it is sufficient to find a set of (not necessarily efficient) solutions that, for each possible image, contains a solution whose image is component-wise at least as good up to a multiplicative constant. Scalarizations are frequently used in approximation methods for multiobjective optimization problems. For example, it has been shown that optimal solutions of the weighted sum scalarization can be used to obtain approximation sets for each instance of each multiobjective minimization problem. However, these approximation results crucially rely on the assumption that all objectives are to be minimized. In fact, it is known that, for the weighted sum scalarization as well as for every other scalarization studied hitherto in view of approximation, even the set of all the optimal solutions of the scalarization does, in general, not constitute an approximation set in the case of maximization problems.

This raises several questions: Are there intrinsic structural differences between minimization and maximization problems with respect to approximation via scalarizations? Does there exist a scalarization such that, in each instance of each maximization problems, optimal solutions of the scalarization constitute an approximation set? What structural properties are necessary for scalarizations to be useful concerning the approximation of multiobjective optimization problems in general?

In this talk, we tackle exactly these questions and study the power of scalarizations for the approximation of multiobjective optimization problems from a general point of view.

Anytime algorithms, which allow a practitioner to trade-off execution time for solution quality, are of particular interest in multi-objective optimization where it is often infeasible to identify all efficient solutions. In this talk, we will discuss a theoretical model that, under some mild assumptions, characterizes the ideal trade-off between execution time and solution quality, measured in terms of hypervolume, of anytime algorithms for bi-objective optimization that collect (efficient) solutions sequentially. We consider this ideal trafe-off under the assumption that the ideal goal of such a sequential algorithm is to maximize the hypervolume each time a solution is collected. We validate our model on benchmark knapsack problems against an oracle that has complete knowledge of the non-dominated set. The empirical results suggest that our theoretical model approximates the behavior of this optimal model quite well. We also show that our model can be used to guide an epsilon-constraint algorithm in order to achieve close to ideal anytime performance.

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