TY - INPR A1 - Shyshkanova, Ganna A1 - Walther, Andrea T1 - Contact Pressure over Doubly Connected Rectangular Domains and Punch Shape Optimization N2 - 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. KW - contact problem KW - shape optimization KW - simple layer potential KW - integral equations Y1 - 2022 ER - TY - JOUR A1 - Kreimeier, Timo A1 - Kuchlbauer, Martina A1 - Liers, Frauke A1 - Stingl, Michael A1 - Walther, Andrea T1 - Towards the Solution of Robust Gas Network Optimization Problems Using the Constrained Active Signature Method N2 - This work studies robust gas network optimization under uncertainties in demand and in the physical parameters. The corresponding optimization problems are nonconvex in node pressures and flows along the pipes. They are thus very difficult to solve for realistic instance sizes. In recent approaches, an adaptive bundle method has been developed, where one solves the occurring adversarial problems via iteratively refined piecewise linear relaxations. These subproblems need to be solved always from scratch using mixed-integer linear programming (MIP). As alternative to the MIP solver, we employ here a nonsmooth optimization approach that allows a warm start strategy such that it can profit from the results obtained for coarser relaxations. We evaluate the approach for realistic gas network topologies and outline possibilities for future research. Y1 - 2022 ER - TY - INPR A1 - Kreimeier, Timo A1 - Sauter, Henning A1 - Streubel, Tom A1 - Tischendorf, Caren A1 - Walther, Andrea T1 - Solving Least-Squares Collocated Differential Algebraic Equations by Successive Abs-Linear Minimization - A Case Study on Gas Network Simulation N2 - This paper studies the numerical simulation of gas networks with regulating elements using differential algebraic equations (DAEs) in combination with least-squares collocation. In contrast to classical collocation methods, more collocation points than degrees of freedom for the collocation polynomials are used. Recently, it has been shown that such a least-squares collocation has a regularizing effect for DAEs, in particular for DAEs with higher index. In each time step of the numerical integration, one has to solve a system of nonlinear equations that is nonsmooth due to the regulating elements in the gas networks. We consider four solvers one of which explicitly exploits the inherent nonsmooth nature. Numerical results are given for three different test cases with increasing complexity illustrating the feasibility of the proposed approach to approximate a solution of the DAE and the advantageous performance of the nonsmooth solver that is based on the concept of abs-linearization. KW - Abs-smooth Algorithmic Differentiation KW - Numerical Integration KW - Simulation of Gas Transportation Networks KW - Nonsmooth Optimization KW - SALMIN KW - Least-Squares Collocation Method KW - Target Values KW - Set Point Values KW - Controlling Regulators and Compressors Y1 - 2021 ER - TY - INPR A1 - Kreimeier, Timo A1 - Pokutta, Sebastian A1 - Walther, Andrea A1 - Woodstock, Zev T1 - On a Frank-Wolfe Approach for Abs-smooth Functions N2 - 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. KW - Frank-Wolfe algorithm KW - Active Signature Method KW - abs-smooth functions KW - nonsmooth optimization KW - convergence rate Y1 - 2022 ER - TY - INPR A1 - Kannan, Aswin A1 - Kreimeier, Timo A1 - Walther, Andrea T1 - On Solving Nonsmooth Retail Portfolio Maximization Problems Using Active Signature Methods Y1 - 2023 ER - TY - INPR A1 - Kreimeier, Timo A1 - Walther, Andrea A1 - Griewank, Andreas T1 - An active signature method for constrained abs-linear minimization N2 - In this paper we consider the solution of optimization tasks with a piecewise linear objective function and piecewise linear constraints. First, we state optimality conditions for that class of problems using the abs-linearization approach and prove that they can be verified in polynomial time. Subsequently, we propose an algorithm called Constrained Active Signature Method that explicitly exploits the piecewise linear structure to solve such optimization problems. Convergence of the algorithm within a finite number of iterations is proven. Numerical results for various testcases including linear complementarity conditions and a bi-level problem illustrate the performance of the new algorithm. Y1 - 2021 ER - TY - JOUR A1 - Shyshkanova, Ganna A1 - Walther, Andrea T1 - Optimization of a punch shape with a doubly connected contact domain N2 - 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. Y1 - 2023 ER - TY - INPR A1 - Graser, Gertrud A1 - Kreimeier, Timo A1 - Walther, Andrea T1 - Solving Linear Generalized Nash Games Using an Active Signature Method N2 - 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. Y1 - 2024 ER -