https://opus4.kobv.de/opus4-trr154OPUS documents
https://opus4.kobv.de/opus4-trr154/index/index/
Mon, 13 Nov 2023 11:20:18 +0100Mon, 13 Nov 2023 11:20:18 +0100On a Tractable Single-Level Reformulation of a Multilevel Model of the European Entry-Exit Gas Market with Market Power
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/527
We propose a framework that allows to quantitatively analyze the interplay of the different agents involved in gas trade and transport in the context of the European entry-exit system. While previous contributions focus on the case of perfectly competitive buyers and sellers of gas, our novel framework considers the mathematically more challenging case of a strategic and monopolistic gas seller. We present a multilevel framework that is suitable to capture the sequential nature of the decisions taken. We then derive sufficient conditions that allow for reformulating the challenging four-level model as a computationally tractable single-level reformulation. We prove the correctness of this reformulation and use it for solving several test instances to illustrate the applicability of our approach.Veronika Grimm; Julia Grübel; Martin Schmidt; Alexandra Schwartz; Ann-Kathrin Wiertz; Gregor Zöttlpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/527Mon, 13 Nov 2023 11:20:18 +0100Solving three-dimensional contact problems for foundation design in green building
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/526
Design of foundations on an elastic base is carried out using the solution of three-dimensional problems of contact interaction. Improving the accuracy of engineering calculations is necessary to ensure economic efficiency and increase energy savings in green building. The problems of indentation of punches with a flat base bounded by doubly connected close to polygonal contact areas are researched in the present work. Small parameter method is used to obtain explicit analytical expressions for the contact pressure distribution and the punch displacement dependence in a simplified form, which is convenient for engineering practice. The found load-displacement dependence satisfies the known inequalities that are valid for an arbitrary contact domain. Also a numerical-analytical method is in consideration. It uses the simple layer potential expansion and successive approximations for the problems accounting roughness of the elastic half-space. Roughness coefficient is considered as a parameter of regularization of the integral equation for the smooth contact problem. The results of both methods coincide with sufficient accuracy.Ganna Shyshkanova; Tetyana Zaytseva; V Zhushman; Ntaliia Levchenko; Olena Korotunovaarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/526Wed, 25 Oct 2023 13:19:46 +0200Decomposition Methods for Time-Dependent Mixed-Integer Nonlinear Optimization Problems on Graphs
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/525
Decomposition can be the method of choice to deal with optimization problems that contain hard to solve model structures or that are of large scale. The main idea is to decompose the problematic aspects of the problem into multiple smaller blocks that can be solved more easily. Here, the challenge is to combine the single pieces to a solution that is not only feasible but maybe even optimal for the original problem. In many cases, this can be done by introducing an iteration that eventually converges to a desired solution.
In this cumulative dissertation, we present several iterative decomposition methods that are tailored to different types of optimization models and use distinct approaches to split up the problems. Our main motivation for this originates from the optimization of gas transport networks, where we encounter partial differential equations as well as discrete control decisions. Additionally, we engage in the related field of district heating network optimization to study the challenges arising from large-scale and fully discretized systems as well as undesirable model features such as, e.g., complementarity constraints. Here, we introduce two temperature mixing models that are well suited for optimization and a number of techniques to speed up the solution process, which are applied in numerical experiments.
As a next step, we develop an iterative time-domain decomposition method that is applied to optimal control problems subject to semilinear hyperbolic systems of partial differential equations. For this, we derive first-order optimality conditions that are then split using a non-overlapping decomposition of the time horizon. We exploit the fact that the resulting systems have a primal interpretation as so-called virtual control problems. We prove the convergence of the iterative method and develop a posteriori error estimates. Later, we extend the scheme to systems of ordinary differential equations with mixed- integer controls by using Pontryagin’s maximum principle. We again show the convergence and conduct a numerical case study.
Moreover, we use a consensus-based version of the classic penalty alternating direction method to solve tailored reformulations of transient gas network problems that allow us to minimize the number of coupling constraints between sub-problems. Here, we utilize the quasi-separable structure of the network to decompose it into sub-networks with more desirable properties. We also discuss different decomposition strategies and test them in a numerical case study. Finally, we present a successive linear relaxation method for mixed-integer nonlinear problems with multivariate Lipschitz continuous nonlinearities. The distinguishing feature of this algorithm is that it exploits no properties of the nonlinearities besides the Lipschitz constants. Therefore, the method is
applicable for problems with non-convex or even non-differentiable constraints. The nonlinearities do not even need to be given in a closed form, which allows us to integrate black-box constraints into the model. We prove that the algorithm converges to an approximate global optimum and we provide a worst-case estimate for the number of iterations. The iterative method is applied to stationary gas transport problems, where implicitly given solutions of the differential equations are modeled via black-box constraints.Richard Krugdoctoralthesishttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/525Tue, 03 Oct 2023 23:01:11 +0200Optimizing the economic dispatch of weakly-connected mini-grids under uncertainty using joint chance constraints
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/524
In this paper, we deal with a renewable-powered mini-grid, connected to an unreliable main grid, in a Joint Chance Constrained (JCC) programming setting.
In many countries with low energy access rates, grid-connected mini-grid system operators contend with four different types of uncertainties: stochastic solar
power and demand forecast errors; absolute uncertain national grid outage onset
times; and outages duration subjected to statistical analysis. These uncertainties
pose new challenges to the classical power system’s operation tasks. Two alternatives to the JCC problem are presented. In particular, we present an Individual
Chance Constraint (ICC) and a purely deterministic dispatch model. The JCC
model has the capability to address all four uncertainties, while the ICC covers only three of them, overlooking the uncertainty about the outage duration.
In contrast, the purely deterministic model completely ignores any uncertain
parameters. We illustrate the three models through a comparison of outcomes
attained from a real mini-grid in Lake Victoria, Tanzania. Results show how the
dispatch is modified across the models to plan the battery and diesel reserves in
the chance-constrained models, with the reserves in the JCC being larger than
in the ICC model. In comparison between all models, we prove that the JCC model offers the most robust results, since it can handle uncertainties about forecasting errors, on the one hand, and grid outages, on the other. The results also
show that the decrease in profits due to the hedging with reserves kept in the
MG is significantly small compared to the high level of reliability reached and
the potential load shedding that could be avoided in the case of an outage.Nesrine Ouanes; Tatiana González Grandón; Holger Heitsch; René Henrionarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/524Mon, 04 Sep 2023 09:35:14 +0200Optimization of a punch shape with a doubly connected contact domain
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/523
The objective is to optimize the pressure distribution under a rigid punch having a doubly connected contact domain close to a circular ring and interacting with an elastic half-space. The required design variable is the punch shape. The functional to be minimized is the root-mean-square deviation of the pressure distribution from some given distribution. An analytical technique is developed for solving the problem for the punches with doubly connected shape, by reducing to a sequence of similar problems for the circular ring punches using expansions of the simple layer potential. The method of expansion in terms of a small parameter is used. The simple layer potential expansion is proposed when mapping a doubly connected integration domain onto a circular ring by transforming the integration variables and transforming the coordinates of the pole of the kernel. As a result, a sequence of similar problems was obtained for a circular ring to determine the functions characterizing the distribution of normal pressure under the punch in the form of a non-circular ring, as well as the normal displacements, from where the optimal punch shape is determined.Ganna Shyshkanova; Andrea Waltherarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/523Tue, 27 Jun 2023 13:12:18 +0200Connections and Reformulations between Robust and Bilevel Optimization
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/522
Robust and bilevel optimization share the common feature that they involve a certain multilevel structure. Hence, although they model something rather different when used in practice, they seem to have a similar mathematical structure. In this paper, we analyze the connections between different types of robust problems (strictly robust problems with and without decision-dependence of their uncertainty sets, min-max-regret problems, and two-stage robust problems) as well as of bilevel problems (optimistic problems, pessimistic problems, and robust bilevel problems). It turns out that bilevel optimization seems to be more general in the sense that for most types of robust problems, one can find proper reformulations as bilevel problems but not necessarily the other way around. We hope that these results pave the way for a stronger connection between the two fields - in particular to use both theory and algorithms from one field in the other and vice versa.Marc Goerigk; Jannis Kurtz; Martin Schmidt; Johannes Thüraufpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/522Tue, 27 Jun 2023 10:55:13 +0200Topological derivative method for control of wave equation on networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/521
The dynamical, boundary optimal control problems
on networks are considered. The domain of definition for the distributed parameter system is given by a graph G. The optimal cost function for control problem is further optimized with respect to the shape and topology of the graph Ω. The small cycle is introduced and the topological derivative of the cost with respect to the size of the cycle is determined. In this way, the singular perturbations of the graph can be analyzed in order to change the topology Ω. The topological derivative method in shape and topology optimization is a new tool which can be used to minimize the shape functionals under the Partial Differential
Equations (PDEs) constraints. The topological derivative is used as well for solution of optimum design problems for graphs. In optimal control problems the topological derivative is used for
optimum design of the domain of integration of the state equation. As an example, optimal control problems are considered on a cross with a small cycle. The state equation is the wave equation
on the graph. The boundary control problem by Neumann
conditions at a boundary vertex is solved for a tracking cost function. The shape functional is given by the optimal value of the control cost. The topological derivative of the shape functional is determined for the steady state model with the size of a cycle ε → 0. Numerical results for a model problem are presented.Martin Gugat; Meizhi Qian; Jan Sokolowskipreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/521Wed, 14 Jun 2023 14:06:06 +0200On Solving Nonsmooth Retail Portfolio Maximization Problems Using Active Signature Methods
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/520
Aswin Kannan; Timo Kreimeier; Andrea Waltherpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/520Mon, 12 Jun 2023 08:38:47 +0200An Algorithmic Framework for Optimal Control of Hybrid Dynamical System with Parabolic PDEs
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/519
We present an algorithmic approach for the computational solution of optimal control problems with hybrid nature governed by linear parabolic PDEs featuring implicit switches. We propose a stepwise reformulation of the original formulation into a more tractable setting via application of methods from disjunctive programming and a time transformation method.
After removal of the implicit switching rule at the cost of the introduction of explicit switching variables and vanishing constraints, the connection of the resulting formulation to problems with equilibrium constraints is established and studied. The previous steps in combination with smoothening and a Moreau-Yosida type penalty approach allow the derivation of necessary first order optimality conditions to characterize candidates for optimality to the original system. Following the discussion of each individual reformulation step, we introduce the algorithmic framework founded on a semismooth Newton method. Finally, we report on computational of the proposed framework.Falk Hante; Christian Kuchlerpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/519Thu, 25 May 2023 15:58:14 +0200Norm-induced Cuts: Optimization with Lipschitzian Black-box Functions
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/518
Optimal control problems usually involve constraints which model physical states and their possible transitions. These are represented by ordinary or partial differential equations (ODEs/PDEs) which add a component of infinite dimension to the problem. In recent literature, one method to simulate such ODEs/PDEs are physics-informed neural networks. Typically, neural networks are highly non-linear which makes their addition to optimization problems challenging. Hence, we leverage their often available Lipschitz property on a compact domain. The respective Lipschitz constants have to be computed only once and are accessible thereafter.
We present a method that, based on this property, iteratively adds cuts involving the violation of the constraints by the current incumbent and the Lipschitz constant. Hereby, the “shape” of a cut depends on the norm used. We prove the correctness of the method by showing that it either returns an optimal solution when terminating or creates a sequence with optimal accumulation points. This is complemented by a discussion about the termination in the infeasible case, as well as an analysis of the problem complexity. For the analysis, we show that the lower and upper iteration bound asymptotically coincide when the relative approximation error goes to zero. In the end, we visualize the method on a small example based on a two-dimensional non-convex optimization problem, as well as stress the necessity of having a globally optimal oracle for the sub-problems by another example.Adrian Göß; Alexander Martin; Sebastian Pokutta; Kartikey Sharmapreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/518Tue, 16 May 2023 08:22:00 +0200Optimal boundary control of the isothermal semilinear Euler equation for gas dynamics on a network
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/512
The analysis and boundary optimal control of the nonlinear transport of gas on a network of pipelines is considered. The evolution of the gas distribution on a given pipe is modeled by an isothermal semilinear compressible Euler system in one space dimension. On the network, solutions satisfying (at nodes) the Kirchhoff flux continuity conditions are shown to exist in a neighborhood of an equilibrium state. The associated nonlinear optimization problem then aims at steering such dynamics to a given target distribution by means of suitable (network) boundary controls while keeping the distribution within given (state) constraints. The existence of local optimal controls is established and a corresponding Karush-Kuhn-Tucker (KKT) stationarity system with an almost surely non-singular Lagrange multiplier is derived.Marcelo Bongarti; Hintermüllerpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/512Wed, 03 May 2023 15:58:27 +0200Max-p optimal boundary control of gas flow
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/510
In the transition to renewable energy sources, hydrogen will potentially play an important role for energy storage. The efficient transport of this gas is possible via pipelines. An understanding of the possibilities to control the gas flow in pipelines is one of the main building blocks towards the optimal use of gas.
For the operation of gas transport networks it is important to take into account the randomness of the consumers’ demand, where often information on the probability distribution is available.
Hence in an efficient optimal control model the corresponding probability should be included and the optimal control should be such that the state that is generated by the optimal control satisfies given state constraints with large probability. We comment on the modelling of gas pipeline flow and the problems of optimal nodal control with random demand, where the aim of the optimization is to determine controls that generate states that satisfy given pressure bounds with large probability. We include the H2 norm of the control as control cost, since this avoids large pressure fluctuations which are harmful in the transport of hydrogen since they can cause
embrittlement of the pipeline metal.Martin Gugat; Michael Schusterconferenceobjecthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/510Fri, 14 Apr 2023 08:28:51 +0200Turnpike Properties for Partially Uncontrollable Systems
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/511
We analyse the turnpike properties for a general, infinite dimensional, linear-quadratic (LQ) optimal control problem, both in the deterministic and in the stochastic case.
The novelty of the paper is twofold. Firstly, it obtains positive turnpike results for
systems that are (partially) uncontrollable. Secondly, it provides turnpike results for averaged control associated to a family of problems that depend on a random parameter, which is the first turnpike type result in the averaged controllability framework.Martin Gugat; Martin Lazararticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/511Fri, 14 Apr 2023 08:27:31 +0200A Turnpike Result for Optimal Boundary Control Problems with the Transport Equation under Uncertainty
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/509
In this paper we analyze the turnpike phenomenon for optimal boundary control problems with a linear transport equation with source term. The convex objective function depends on
the boundary traces of the transport equation and is strictly convex with respect to the boundary control. We show an integral turnpike result for an optimal Dirichlet boundary control problem in the sense that if the time horizon goes to infinity, then the dynamic optimal control converges to
the corresponding steady state optimal control.
The novelty of this work is two-sided. On the one hand, even if turnpike results for this kind of optimal boundary control problem already exist, we present a new direct proof without using adjoint calculus that leads to sharper estimates. On the other hand we consider uncertainty in
the initial data and/or in the source term. We show that the integral turnpike result also holds considering uncertainty. Throughout the paper we use numerical examples to illustrate the results.Michael Schuster; Noboru Sakamotopreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/509Tue, 14 Mar 2023 12:14:12 +0100Observer-based data assimilation for barotropic gas transport using distributed measurements
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/508
We consider a state estimation problem for gas pipeline flow modeled by the one-dimensional barotropic Euler equations. In order to reconstruct the system state, we construct an observer system of Luenberger type based on distributed measurements of one state variable. First, we show the existence of Lipschitz-continuous semi-global solutions of the observer system and of the original system for initial and boundary data satisfying smallness and compatibility conditions for a single pipe and for general networks. Second, based on an extension of the relative energy method we prove that the state of the observer system converges exponentially in the long time limit towards the original system state. We show this for a single pipe and for star-shaped networks.Jan Giesselmann; Martin Gugat; Teresa Kunkelpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/508Fri, 10 Mar 2023 08:53:51 +0100The economics of global green ammonia trade – "Shipping Australian wind and sunshine to Germany"
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/507
This paper contributes to understanding the transformation of global energy trade to green energy carriers, focusing on green ammonia as the foreseeable first green hydrogen carrier. We provide a comprehensive overview of today's ammonia trade and assess scaling options for the trade of green ammonia. To that aim, we develop an optimization model for the integrated assessment of the green ammonia value chain that covers all steps from green ammonia production in an exporting country, up to delivery to a harbor in an importing country. The model endogenously chooses among different technology options and determines cost minimal operation. In a case study, we apply the model to the large-scale import of ammonia from Australia to Germany in a scenario for 2030. The results show that green ammonia can reach cost parity with gray ammonia even for moderate gas prices (but not necessarily with blue ammonia) if CO2 prices are high enough. We also provide a sensitivity analysis with respect to the interest rate and other key technical and economic parameters and show that cracking ammonia to provide pure hydrogen comes at a 45 % cost markup per MWh at the destination.Jonas Egerer; Veronika Grimm; Kiana Niazmand; Philipp Rungearticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/507Tue, 14 Feb 2023 11:09:19 +0100Gas Transport Network Optimization: Mixed-Integer Nonlinear Models
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/505
Although modern societies strive towards energy systems that are entirely based on renewable energy carriers, natural gas is still one of the most important energy sources. This became even more obvious in Europe with Russia's 2022 war against the Ukraine and the resulting stop of gas supplies from Russia. Besides that it is very important to use this scarce resource efficiently. To this end, it is also of significant relevance that its transport is organized in the most efficient, i.e., cost- or energy-efficient, way. The corresponding mathematical optimization models have gained a lot of attention in the last decades in different optimization communities. These models are highly nonlinear mixed-integer problems that are constrained by algebraic constraints and partial differential equations (PDEs), which usually leads to models that are not tractable. Hence, simplifications have to be made and in this chapter, we present a commonly accepted finite-dimensional stationary model, i.e., a model in which the steady-state solutions of the PDEs are approximated with algebraic constraints. For more details about the involved PDEs and the treatment of transient descriptions we refer to Hante and Schmidt (2023). The presented finite-dimensional as well as mixed-integer nonlinear and nonconvex model is still highly challenging if it needs to be solved for real-world gas transport networks. Hence, we also review some classic solution approaches from the literature.Falk M. Hante; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/505Fri, 20 Jan 2023 10:26:35 +0100Gas Transport Network Optimization: PDE-Constrained Models
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/504
The optimal control of gas transport networks was and still is a very important topic for modern economies and societies. Accordingly, a lot of research has been carried out on this topic during the last years and decades. Besides mixed-integer aspects in gas transport network optimization, one of the main challenges is that a physically and technically detailed modeling of transient gas dynamics leads to theoretically and computationally highly demanding models involving nonlinear partial differential equations (PDEs). For further background on the application, historical notes and a detailed discussion of mixed-integer aspects for stationary descriptions we refer to Hante and Schmidt (2023). In this chapter, we focus on the most common modeling approaches concerning transient descriptions, point out the challenges, and summarize important contributions concerning the optimization of the most relevant control parameters for this particular class of problems.Falk M. Hante; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/504Fri, 20 Jan 2023 10:25:58 +0100A Consensus-Based Alternating Direction Method for Mixed-Integer and PDE-Constrained Gas Transport Problems
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/503
We consider dynamic gas transport optimization problems, which lead to large-scale and nonconvex mixed-integer nonlinear optimization problems (MINLPs) on graphs. Usually, the resulting instances are too challenging to be solved by state-of-the-art MINLP solvers. In this paper, we use graph decompositions to obtain multiple optimization problems on smaller blocks, which can be solved in parallel and which may result in simpler classes of optimization problems since not every block necessarily contains mixed-integer or nonlinear aspects. For achieving feasibility at the interfaces of the several blocks, we employ a tailored consensus-based penalty alternating direction method. Our numerical results show that such decomposition techniques can outperform the baseline approach of just solving the overall MINLP from scratch. However, a complete answer to the question of how to decompose MINLPs on graphs in dependence of the given model is still an open topic for future research.Richard Krug; Günter Leugering; Alexander Martin; Martin Schmidt; Dieter Weningerpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/503Mon, 24 Oct 2022 09:54:12 +0200An Exact Method for Nonlinear Network Flow Interdiction Problems
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/502
We study network flow interdiction problems with nonlinear and nonconvex flow models. The resulting model is a max-min bilevel optimization problem in which the follower's problem is nonlinear and nonconvex. In this game, the leader attacks a limited number of arcs with the goal to maximize the load shed and the follower aims at minimizing the load shed by solving a transport problem in the interdicted network. We develop an exact algorithm consisting of lower and upper bounding schemes that computes an optimal interdiction under the assumption that the interdicted network remains weakly connected. The main challenge consists of computing valid upper bounds for the maximal load shed, whereas lower bounds can directly be derived from the follower's problem. To compute an upper bound, we propose solving a specific bilevel problem, which is derived from restricting the flexibility of the follower when adjusting the load flow. This bilevel problem still has a nonlinear and nonconvex follower's problem, for which we then prove necessary and sufficient optimality conditions. Consequently, we obtain equivalent single-level reformulations of the specific bilevel model to compute upper bounds. Our numerical results show the applicability of this exact approach using the example of gas networks.Martin Schmidt; Johannes Thüraufpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/502Fri, 21 Oct 2022 07:58:49 +0200Contact Pressure over Doubly Connected Rectangular Domains and Punch Shape Optimization
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/501
Contact problems arise in a variety of industrial processes, engineering and biomechanical systems. 3-D contact problem for a rigid punch with a doubly connected base bounded by the lines close to rectangles is in consideration. An analytic-numerical technique is developed for its solving. The problem contains Fredholm integral equations of the first kind, which are transformed into the second kind by means of regularization. Using the simple layer potential expansion, the kernels of the integrals are presented in the form of expansions in the powers of the polar radius. The difference between the values of the desired function at different points and the subsequent interpolation of the terms are proposed to smooth the kernels and eliminate singularities. The integral equations are reduced to one-dimension and then solved using quadrature formulas. Subsequently a punch shape is taken as a desired function, and as a minimizing functional is considered the root-mean-square deviation of the pressure distribution arising under the punch from some optimal distribution. In this case, the values of the total forces and moments applied to the punch are assumed to be given, which leads to restrictions imposed on the distributions by the equilibrium conditions. The normal displacements are determined which arising under the action of the found contact pressure on the elastic half-space. The desired punch shape is found using the simple layer potential. A solution to the problem is obtained for the punch with the doubly connected base bounded by lines close to rectangles.Ganna Shyshkanova; Andrea Waltherpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/501Wed, 28 Sep 2022 08:36:08 +0200On a Frank-Wolfe Approach for Abs-smooth Functions
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/499
We propose an algorithm which appears to be the first bridge between the fields of conditional gradient methods and abs-smooth optimization. Our nonsmooth nonconvex problem setting is motivated by machine learning, since the broad class of abs-smooth functions includes, for instance, the squared $\ell_2$-error of a neural network with ReLU or hinge Loss activation. To overcome the nonsmoothness in our problem, we propose a generalization to the traditional Frank-Wolfe gap and prove that first-order minimality is achieved when it vanishes. We derive a convergence rate for our algorithm which is identical to the smooth case. Although our algorithm necessitates the solution of a subproblem which is more challenging than the smooth case, we provide an efficient numerical method for its partial solution, and we identify several applications where our approach fully solves the subproblem. Numerical and theoretical convergence is demonstrated, yielding several conjectures.Timo Kreimeier; Sebastian Pokutta; Andrea Walther; Zev Woodstockpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/499Sat, 03 Sep 2022 22:43:38 +0200A Successive Linear Relaxation Method for MINLPs with Multivariate Lipschitz Continuous Nonlinearities
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/498
We present a novel method for mixed-integer optimization problems with multivariate and Lipschitz continuous nonlinearities. In particular, we do not assume that the nonlinear constraints are explicitly given but that we can only evaluate them and that we know their global Lipschitz constants. The algorithm is a successive linear relaxation method in which we alternate between solving a master problem, which is a mixed-integer linear relaxation of the original problem, and a subproblem, which is designed to tighten the linear relaxation of the next master problem by using the Lipschitz information about the respective functions. By doing so, we follow the ideas of Schmidt et al. (2018, 2021) and improve the tackling of multivariate constraints. Although multivariate nonlinearities obviously increase modeling capabilities, their incorporation also significantly increases the computational burden of the proposed algorithm. We prove the correctness of our method and also derive a worst-case iteration bound. Finally, we show the generality of the addressed problem class and the proposed method by illustrating that both bilevel optimization problems with nonconvex and quadratic lower levels as well as nonlinear and mixed-integer models of gas transport can be tackled by our method. We provide the necessary theory for both applications and briefly illustrate the outcomes of the new method when applied to these two problems.Julia Grübel; Richard Krug; Martin Schmidt; Winnifried Wollnerpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/498Fri, 12 Aug 2022 17:33:09 +0200A Penalty Branch-and-Bound Method for Mixed-Integer Quadratic Bilevel Problems
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/497
We propose an algorithm for solving bilevel problems with mixed-integer convex-quadratic upper level as well as convex-quadratic and continuous lower level. The method is based on a classic branch-and-bound procedure, where branching is performed on the integer constraints and on the complementarity constraints resulting from the KKT reformulation of the lower-level problem. However, instead of branching on constraints as usual, suitably chosen penalty terms are added to the objective function in order to create new subproblems in the tree. We prove the correctness of the method and present its applicability by some first numerical results.Andreas Horländer; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/497Sun, 17 Jul 2022 11:01:53 +0200Data-driven Distributionally Robust Optimization over Time
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/496
Stochastic Optimization (SO) is a classical approach for optimization under uncertainty that typically requires knowledge about the probability distribution of uncertain parameters. As the latter is often unknown, Distributionally Robust Optimization (DRO) provides a strong alternative that determines the best guaranteed solution over a set of distributions (ambiguity set). In this work, we present an approach for DRO over time that uses online learning and scenario observations arriving as a data stream to learn more about the uncertainty. Our robust solutions adapt over time and reduce the cost of protection with shrinking ambiguity. For various kinds of ambiguity sets, the robust solutions converge to the SO solution. Our algorithm achieves the optimization and learning goals without solving the DRO problem exactly at any step. We also provide a regret bound for the quality of the online strategy which converges at a rate of $ O(\log T / \sqrt{T})$, where $T$ is the number of iterations. Furthermore, we illustrate the effectiveness of our procedure by numerical experiments on mixed-integer optimization instances from popular benchmark libraries and give practical examples stemming from telecommunications and routing. Our algorithm is able to solve the DRO over time problem significantly faster than standard reformulations.Kevin-Martin Aigner; Andreas Bärmann; Kristin Braun; Frauke Liers; Sebastian Pokutta; Oskar Schneider; Kartikey Sharma; Sebastian Tschuppikpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/496Mon, 11 Jul 2022 14:33:30 +0200