Gertrud
Graser
Lösen von Multi-Leader-Follower Spielen über nichtglatte Nash Equilibrium Probleme
Forschungsseminar Algorithmische Optimierung  📅
Institute
Head
Falk Hante
and
Andrea Walther
Usual time
Thursdays 15:15
Usual venue
Rudower Chaussee 25, Raum 2.417
Number of talks
10
July 2024
June 2024
Tuan Thanh
Truong
AG H. Gottschalk, TU Berlin
and
Miguel
Matamouros Resende Frade de Campos
AG H. Gottschalk, TU Berlin
Shape definition and shape optimization with invertible neural networks
Vladimir
Shikmann
Technische Universität Chemnitz
Cardinality-constrained optimization problems in general position and beyond
Abstract.
We study cardinality-constrained optimization problems (CCOP) in general position, i.e. those optimization-related properties that are fulfilled for a dense and open subset of their defining functions. We show that the well-known cardinality-constrained linear independence constraint qualification (CC-LICQ) is generic in this sense. For M-stationary points we define nondegeneracy and show that it is a generic property too. In particular, the sparsity constraint turns out to be active at all minimizers of a generic CCOP. Moreover, we describe the global structure of CCOP in the sense of Morse theory, emphasizing the strength of the generic approach. Here, we prove that multiple cells need to be attached, each of dimension coinciding with the proposed M-index of nondegenerate M-stationary points. Beyond this generic viewpoint, we study singularities of CCOP. For that, the relation between nondegeneracy and strong stability in the sense of Kojima is examined. We show that nondegeneracy implies the latter, while the reverse implication is in general not true. To fill the gap, we fully characterize the strong stability of M-stationary points under CC-LICQ by first- and second-order information of CCOP defining functions. Finally, we compare nondegeneracy and strong stability of M-stationary points with second-order sufficient conditions recently introduced in the literature.
May 2024
Kathrin
Flaßkamp
Universität des Saarlandes
Optimal Path Planning in Stereotactic Neurosurgery
April 2024
Markus
Herrmann-Wicklmayr
Universität des Saarlandes
Accelerating Multi-Objective Model Predictive Control Using High-Order Sensitivity Information
February 2024
Vladimir
Shikhman
TU Chemnitz
Moritz
Link
Universität Konstanz
Multiobjective mixed-integer nonlinear optimization with application to energy supply networks
Abstract.
In light of the ongoing developments in the climate crisis, it is necessary to consider factors beyond the sole economic perspective in energy supply network planning. This gives rise to a classical multiobjective optimization problem involving conflicting objective functions. The presence of choices in the model, coupled with the consideration of stationary flow equations, results in an optimization problem with a mixed-integer nonlinear structure. Following a short introduction of some model-specific details and arising challenges, we present a novel method for computing an enclosure of the nondominated set of such problems. We prove finite convergence and demonstrate its effectiveness through numerical experiments. We conclude by offering a preview of ongoing extensions of this work.
January 2024
Bennet
Gebken
Universität Paderborn
First- and second-order descent methods for nonsmooth optimization based on deterministic gradient sampling
Abstract.
In nonsmooth optimization, it is well known that the classical gradient is not a suitable object for describing the local behavior of a function. Instead, generalized derivatives from nonsmooth analysis, like the Clarke subdifferential, have to be employed. While in theory, the Clarke subdifferential inherits many useful properties from the classical gradient, there is a large discrepancy in practice: It is unstable, and for a general locally Lipschitz continuous function, it is impossible to compute. Thus, in practice, the Clarke subdifferential has to be approximated. A simple strategy to achieve this, known as gradient sampling, is based on approximating it by taking the convex hull of classical gradients evaluated at smooth points from a small neighborhood of a given point. In this talk, I will present two descent methods for nonsmooth optimization that are based on this idea. However, in contrast to the standard gradient sampling approach, where the gradients are sampled randomly, both methods will be deterministic. I will begin with a first-order method, where new gradients are computed using a bisection subroutine. Afterwards, I will demonstrate how the gradient sampling methodology can be generalized to second-order derivates by sampling Hessian matrices in addition to gradients. This will lead to a second-order descent method that, at least in numerical experiments, shows a high speed of convergence.
December 2023
Gabriele
Eichfelder
Fachgebiet Mathematische Methoden des Operations Research, TU Ilmenau
Multiobjective Mixed Integer Programming
Abstract.
Multiobjective mixed integer nonlinear optimization refers to mathematical programming problems where more than one nonlinear objective function needs to be optimized simultaneously and some of the variables are constrained to take integer values. In this talk, we give a short introduction to the basic concepts of multiobjective optimization. We give insights why the famous approach of scalarization might not be an appropriate method to solve these problems. Instead, we present two procedures to solve the problems directly. The first is a branch-and-bound method based on the use of properly defined lower bounds. We do not simply rely on convex relaxations, but we built linear outer approximations of the image set in an adaptive way. The second method is tailored for convex objective functions and is purely based on the criterion space. It uses ingredients from the well-known outer approximation algorithm from single-objective mixed-integer optimization and combines them with strategies to generate enclosures of nondominated sets by iteratively improving approximations. For both algorithms, we are able to guarantee correctness in terms of detecting the nondominated set of multiobjective mixed integer problems according to a prescribed precision.
November 2023
Sonja
Steffensen
Institut für Geometrie und Praktische Mathematik, RWTH Aachen
On Multilevel Game Theory and its Applications