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Thu, 16 May 2024 23:40:42 +0200Thu, 16 May 2024 23:40:42 +0200Outer approximation for generalized convex mixed-integer nonlinear robust optimization problems
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/551
We consider nonlinear robust optimization problems with mixed-integer decisions as well as nonconvexities. In detail, we consider cases where objective and constraint functions can be nonsmooth and generalized convex, i.e., f°-quasiconvex or f°-pseudoconvex.
We propose an algorithm for such robust optimization problems that does not require a certain structure of the adversarial problem but only requires that approximate worst cases are available. As a result, our algorithm finds a robust optimal solution up to a tolerance. Our method integrates a bundle method into an outer approximation approach where the bundle method is used for the arising continuous subproblems. We rely on methods from the literature, namely a bundle method for nonlinear and nonconvex robust optimization problems and outer approximation approaches for quasiconvex settings. Our contribution is to combine them to one convergent robust optimization method that can cope with inexactness of worst-case evaluations.
Further, we propose the gas transport under uncertainties as a relevant application and demonstrate that generalized convexity is fulfilled for a type of a network structure.Martina Kuchlbauerpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/551Thu, 16 May 2024 23:40:42 +0200Adjustable Robust Nonlinear Network Design under Demand Uncertainties
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/550
We study network design problems for nonlinear and nonconvex flow models under demand uncertainties. To this end, we apply the concept of adjustable robust optimization to compute a network design that admits a feasible transport for all, possibly infinitely many, demand scenarios within a given uncertainty set. For solving the corresponding adjustable robust mixed-integer nonlinear optimization problem, we show that a given network design is robust feasible, i.e., it admits a feasible transport for all demand uncertainties, if and only if a finite number of worst-case demand scenarios can be routed through the network. We compute these worst-case scenarios by solving polynomially many nonlinear optimization problems. Embedding this result for robust feasibility in an adversarial approach leads to an exact algorithm that computes an optimal robust network design in a finite number of iterations. Since all of the results are valid for general potential-based flows, the approach can be applied to different utility networks such as gas, hydrogen, or water networks. We finally demonstrate the applicability of the method by computing robust gas networks that are protected from future demand fluctuations.Johannes Thürauf; Julia Grübel; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/550Sat, 06 Apr 2024 22:41:05 +0200Adaptive hp-Polynomial Based Sparse Grid Collocation Algorithms for Piecewise Smooth Functions with Kinks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/549
High-dimensional interpolation problems appear in various applications of uncertainty quantification, stochastic optimization and machine learning. Such problems are computationally expensive and request the use of adaptive grid generation strategies like anisotropic sparse grids to mitigate the curse of dimensionality. However, it is well known that the standard dimension-adaptive sparse grid method converges very slowly or even fails in the case of non-smooth functions. For piecewise smooth functions with kinks, we construct two novel hp-adaptive sparse grid collocation algorithms that combine low-order basis functions with local support in parts of the domain with less regularity and variable-order basis functions elsewhere. Spatial refinement is realized by means of a hierarchical multivariate knot tree which allows the construction of localised hierarchical basis functions with varying order. Hierarchical surplus is used as an error indicator to automatically detect the non-smooth region and adaptively refine the collocation points there. The local polynomial degrees are optionally selected by a greedy approach or a kink detection procedure. Three numerical benchmark examples with different dimensions are discussed and comparison with locally linear and highest degree basis functions are given to show the efficiency and accuracy of the proposed methods.Hendrik Wilka; Jens Langpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/549Thu, 04 Apr 2024 19:13:31 +0200A Stiff MOL Boundary Control Problem for the 1D Heat Equation with Exact Discrete Solution
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/547
Method-of-lines discretizations are demanding test problems for stiff inte-
gration methods. However, for PDE problems with known analytic solution
the presence of space discretization errors or the need to use codes to compute
reference solutions may limit the validity of numerical test results. To over-
come these drawbacks we present in this short note a simple test problem with
boundary control, a situation where one-step methods may suffer from order
reduction. We derive exact formulas for the solution of an optimal boundary
control problem governed by a one-dimensional discrete heat equation and an
objective function that measures the distance of the final state from the target
and the control costs. This analytical setting is used to compare the numeri-
cally observed convergence orders for selected implicit Runge-Kutta and Peer
two-step methods of classical order four which are suitable for optimal control
problems.Jens Lang; Bernhard A. Schmittarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/547Mon, 25 Mar 2024 12:09:02 +0100Physics-Informed Neural Networks: A Case Study for Gas Transport Problems
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/546
Physics informed neural networks have been recently proposed and offer a new promising method to solve differential equations. They have been adapted to many more scenarios and different variations of the original method have been proposed. In this case study we review many of these variations. We focus on variants that can compensate for imbalances in the loss function and perform a comprehensive numerical comparison of these variants with application to gas transport problems. Our case study includes different formulations of the loss function, different algorithmic loss balancing methods, different optimization schemes and different numbers of parameters and sampling points. We conclude that the original PINN approach with specifically chosen constant weights in the loss function gives the best results in our tests. These weights have been obtained by a computationally expensive random-search scheme. We further conclude for our test case that loss balancing methods which were developed for other differential equations have no benefit for gas transport problems, that the control volume physics informed formulation has no benefit against the initial formulation and that the best optimization strategy is the L-BFGS method.Erik Laurin Strelow; Alf Gerisch; Jens Lang; Marc E. Pfetscharticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/546Mon, 25 Mar 2024 12:05:36 +0100Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/545
This paper is concerned with the construction and convergence analysis
of novel implicit Peer triplets of two-step nature with four stages for nonlinear
ODE constrained optimal control problems. We combine the property of superconvergence
of some standard Peer method for inner grid points with carefully
designed starting and end methods to achieve order four for the state variables
and order three for the adjoint variables in a first-discretize-then-optimize approach
together with A-stability. The notion triplets emphasizes that these
three different Peer methods have to satisfy additional matching conditions.
Four such Peer triplets of practical interest are constructed. Also as a benchmark
method, the well-known backward differentiation formula BDF4, which is
only A(73.35)-stable, is extended to a special Peer triplet to supply an adjoint
consistent method of higher order and BDF type with equidistant nodes. Within
the class of Peer triplets, we found a diagonally implicit A(84)-stable method
with nodes symmetric in [0,1] to a common center that performs equally well.
Numerical tests with three well established optimal control problems confirm
the theoretical findings also concerning A-stability.Jens Lang; Bernhard A. Schmittarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/545Thu, 21 Mar 2024 21:02:23 +0100Gas Network Modeling: An Overview (Extended English Version)
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/543
With this overview we want to provide a compilation of different models for
the description of gas flow in networks in order to facilitate the introduction
to the topic. Special attention is paid to the hierarchical structure inherent
to the modeling, and the detailed description of individual components such
as valves and compressors. Also included are network model classes based
on purely algebraic relations, and energy-based port-Hamiltonian models. A
short overview of basic numerical methods and concepts for the treatment
of hyperbolic balance equations is also given. We do not claim completeness
and refer in many places to the existing literature.Pia Domschke; Jan Giesselmann; Jens Lang; Tobias Breiten; Volker Mehrmann; Riccardo Morandin; Benjamin Hiller; Caren Tischendorfpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/543Thu, 21 Mar 2024 20:55:41 +0100Implicit Peer Triplets in Gradient-Based Solution Algorithms for ODE Constrained Optimal Control
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/548
It is common practice to apply gradient-based optimization algorithms to
numerically solve large-scale ODE constrained optimal control problems. Gradients
of the objective function are most efficiently computed by approximate
adjoint variables. High accuracy with moderate computing time can be achieved
by such time integration methods that satisfy a sufficiently large number of adjoint
order conditions and supply gradients with higher orders of consistency. In
this paper, we upgrade our former implicit two-step Peer triplets constructed in
[Algorithms, 15:310, 2022] to meet those new requirements. Since Peer methods
use several stages of the same high stage order, a decisive advantage is their lack
of order reduction as for semi-discretized PDE problems with boundary control.
Additional order conditions for the control and certain positivity requirements
now intensify the demands on the Peer triplet. We discuss the construction of
4-stage methods with order pairs (4,3) and (3,3) in detail and provide three
Peer triplets of practical interest. We prove convergence for s-stage methods,
for instance, order s for the state variables even if the adjoint method and the
control satisfy the conditions for order s-1, only. Numerical tests show the
expected order of convergence for the new Peer triplets.Jens Lang; Bernhard A. Schmittpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/548Thu, 21 Mar 2024 20:53:30 +0100Solving Linear Generalized Nash Games Using an Active Signature Method
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/542
We propose a method to solve linear generalized Nash equilibrium problems (LGNEPs). For this purpose, a reformulation of the LGNEPs as piecewise linear problems is considered. This requires the calculation of all vertices for a special kind of unbounded convex polyhedra. Then the active signature method for constrained abs-linear problems can be used to determine the Nash equilibria. We analyse the computational effort for the resulting solution procedure. This includes also the verification of suitable optimality conditions. Finally, we present and analyse numerical results for some test problems.Gertrud Graser; Timo Kreimeier; Andrea Waltherpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/542Thu, 29 Feb 2024 15:15:52 +0100On the convergence of the regularized entropy-based moment method for kinetic equations
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/536
The entropy-based moment method is a well-known discretization for the velocity variable in kinetic equations which has many desirable theoretical properties but is difficult to implement with high-order numerical methods. The regularized entropy-based moment method was recently introduced to remove one of the main challenges in the implementation of the entropy-based moment method, namely the requirement of the realizability of the numerical solution. In this work we use the method of relative entropy to prove the convergence of the regularized method to the original method as the regularization parameter goes to zero and give convergence rates. Our main assumptions are the boundedness of the velocity domain and that the original moment solution is Lipschitz continuous in space and bounded away from the boundary of realizability. We provide results from numerical simulations showing that the convergence rates we prove are optimal.Graham Alldredge; Martin Frank; Jan Giesselmannpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/536Tue, 13 Feb 2024 17:44:42 +0100A posteriori error analysis of a positivity preserving scheme for the power-law diffusion Keller-Segel model
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/540
We study a finite volume scheme approximating a parabolic-elliptic Keller-Segel system with power law diffusion with exponent γ∈[1,3] and periodic boundary conditions. We derive conditional a posteriori bounds for the error measured in the L∞(0,T;H1(Ω)) norm for the chemoattractant and by a quasi-norm-like quantity for the density. These results are based on stability estimates and suitable conforming reconstructions of the numerical solution. We perform numerical experiments showing that our error bounds are linear in mesh width and elucidating the behaviour of the error estimator under changes of γ.Jan Giesselmann; Niklas Kolbepreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/540Tue, 13 Feb 2024 17:44:07 +0100A posteriori error control for a Discontinuous Galerkin approximation of a Keller-Segel model
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/541
We provide a posteriori error estimates for a discontinuous Galerkin scheme for the parabolic-elliptic Keller-Segel system in 2 or 3 space dimensions. The estimates are conditional, in the sense that an a posteriori computable quantity needs to be small enough - which can be ensured by mesh refinement - and optimal in the sense that the error estimator decays with the same order as the error under mesh refinement. A specific feature of our error estimator is that it can be used to prove existence of a weak solution up to a certain time based on numerical results.Jan Giesselmann; Kiwoong Kwonpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/541Tue, 13 Feb 2024 17:43:18 +0100Theory of shifts, shocks, and the intimate connections to L2-type a posteriori error analysis of numerical schemes for hyperbolic problems
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/539
In this paper, we develop reliable a posteriori error estimates for numerical approximations of scalar hyperbolic conservation laws in one space dimension.
Our methods have no inherent small-data limitations and are a step towards error control of numerical schemes for systems. We are careful not to appeal to the Kruzhkov theory for scalar conservation laws. Instead, we derive novel quantitative stability estimates that extend the theory of shifts, and in particular, the framework for proving stability first developed by the second author and Vasseur. This is the first time this methodology has been used for quantitative estimates.
We work entirely within the context of the theory of shifts and a-contraction, techniques which adapt well to systems. In fact, the stability framework by the second author and Vasseur has itself recently been pushed to systems [Chen-Krupa-Vasseur. Uniqueness and weak-BV stability for 2×2 conservation laws. Arch. Ration. Mech. Anal., 246(1):299--332, 2022].
Our theoretical findings are complemented by a numerical implementation in MATLAB and numerical experiments.Jan Giesselmann; Sam Krupapreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/539Tue, 13 Feb 2024 17:42:30 +0100Regularity and long time behavior of a doubly nonlinear parabolic problem and its discretization
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/538
We study a doubly nonlinear parabolic problem arising in the modeling of gas transport in pipelines. Using convexity arguments and relative entropy estimates we show uniform bounds and exponential stability of discrete approximations obtained by a finite element method and implicit time stepping. Due to convergence of the approximations to weak solutions of the problem, our results also imply regularity, uniqueness, and long time stability of weak solutions of the continuous problem.Herbert Egger; Jan Giesselmannpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/538Tue, 13 Feb 2024 17:41:44 +0100An Observer for pipeline flow with hydrogen blending in gas networks: exponential synchronization
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/537
We consider a state estimation problem for gas flows in pipeline networks where hydrogen is blended into the natural gas. The flow is modeled by the quasi-linear isothermal Euler equations coupled to an advection equation on a graph. The flow through the vertices where the pipes are connected is governed by algebraic node conditions. The state is approximated by an observer system that uses nodal measurements. We prove that the state of the observer system converges to the original system state exponentially fast in the L2-norm if the measurements are exact. If measurement errors are present we show that the observer state approximates the original system state up to an error that is proportional to the maximal measurement error. The proof of the synchronization result uses Lyapunov functions with exponential weights.Martin Gugat; Jan Giesselmannpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/537Tue, 13 Feb 2024 17:40:51 +0100Single and joint chance-constrained optimization with continuous distributions
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/532
The input parameters of an optimization problem are often affected by uncertainties. Chance constraints are a common way to model stochastic uncertainties in the constraints. Typically, algorithms for solving chance-constrained problems require convex functions or discrete probability distributions. In this work, we go one step further and allow non-convexities as well as continuous distributions. We propose a gradient-based approach to approximately solve joint chance-constrained models. We approximate the original problem by smoothing indicator functions. Then, the smoothed chance constraints are relaxed by penalizing their violation in the objective function. The approximation problem is solved with the Continuous Stochastic Gradient method that is an enhanced version of the stochastic gradient descent and has recently been introduced in the literature. We present a convergence theory for the smoothing and penalty approximations. Under very mild assumptions, our approach is applicable to a wide range of chance-constrained optimization problems. As an example, we illustrate its computational efficiency on difficult practical problems arising in the operation of gas networks. The numerical experiments demonstrate that the approach quickly finds nearly feasible solutions for joint chance-constrained problems with non-convex constraint functions and continuous distributions, even for realistically-sized instances.Daniela Bernhard; Frauke Liers; Michael Stinglpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/532Tue, 13 Feb 2024 12:30:20 +0100Optimality conditions in control problems with random state constraints in probabilistic or almost-sure form
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/529
In this paper, we discuss optimality conditions for optimization problems {involving} random state constraints, which are modeled in probabilistic or almost sure form. While the latter can be understood as the limiting case of the former, the derivation of optimality conditions requires substantially different approaches. We apply them to a linear elliptic partial differential equation (PDE) with random inputs.
In the probabilistic case, we rely on the spherical-radial decomposition of Gaussian random vectors in order to formulate fully explicit optimality conditions involving a spherical integral. In the almost sure case, we derive optimality conditions and compare them to a model based on robust constraints with respect to the (compact) support of the given distribution.Caroline Geiersbach; René Henrionarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/529Mon, 15 Jan 2024 17:47:35 +0100Numerical solution of an optimal control problem with probabilistic and almost sure state constraints
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/530
We consider the optimal control of a PDE with random source term subject to probabilistic or almost sure state constraints. In the main theoretical result, we provide an exact formula for the Clarke subdifferential of the probability function without a restrictive assumption made in an earlier paper. The focus of the paper is on numerical solution algorithms. As for probabilistic constraints, we apply the method of spherical radial decomposition. Almost sure constraints are dealt with a Moreau--Yosida smoothing of the constraint function accompanied by Monte Carlo sampling of the given distribution or its support or even just the boundary of its support. Moreover, one can understand the almost sure constraint as a probabilistic constraint with safety level one which offers yet another perspective. Finally, robust optimization can be applied efficiently when the support is sufficiently simple. A comparative study of these five different methodologies is carried out and illustrated.Caroline Geiersbach; René Henrion; Pedro Pérez-Arosarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/530Mon, 15 Jan 2024 17:43:42 +0100Indirect methods for optimal control of hybrid PDE-dynamical / switching systems using relaxation
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/528
We propose a novel algorithmic approach to computationally solve optimal control problems governed by linear evolution-type PDEs including a state-dependent control-regime switching mechanism. We introduce an equivalent mixed-integer formulation featuring vanishing constraints arising by methods of disjunctive programming. We embed the problem into the class of equilibrium constraints by introduction of an additional slack variable. Based on theoretical results associated with Sum-Up-Rounding strategies, we proceed with the solution of the related relaxed formulation by an indirect approach. In order to obtain a computationally tractable optimality system, we apply a Moreau-Yosida type penalty approach of the vanishing constraints. After the theoretical discussion, we introduce and exert the algorithmic framework founded on a semismooth Newton method. Finally, we communicate computational experiments based on our approach.Falk Hante; Christian Kuchlerpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/528Mon, 11 Dec 2023 12:54:58 +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 +0200