Humboldt-Universität zu Berlin
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- Gas networks (2)
- Mixed-integer nonlinear optimization (2)
- spheric-radial decomposition (2)
- Abs-smooth Algorithmic Differentiation (1)
- Active Signature Method (1)
- Closed-loop stability (1)
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- Euler-Gleichungen, isotherme Euler-Gleichungen, Modellhierarchie, Netzelemente (1)
We present a feedback scheme for non-cooperative dynamic games and investigate its stabilizing properties. The dynamic games are modeled as generalized Nash equilibrium problems (GNEP), in which the shared constraint consists of linear time-discrete dynamic equations (e.g., sampled from a partial or ordinary differential equation), which are jointly controlled by the players’ actions. Further, the individual objectives of the players are interdependent and defined over a fixed time horizon. The feedback law is synthesized by moving-horizon model predictive control (MPC). We investigate the asymptotic stability of the resulting closed-loop dynamics. To this end, we introduce α-quasi GNEPs, a family of auxiliary problems based on a modification of the Nikaido–Isoda function, which approximate the original games. Basing the MPC scheme on these auxiliary problems, we derive conditions on the players’ objectives, which guarantee asymptotic stability of the closed-loop if stabilizing end constraints are enforced. This analysis is based on showing that the associated optimal-value function is a Lyapunov function. Additionally, we identify a suitable Lyapunov function for the MPC scheme based on the original GNEP, whose solution fulfills the stabilizing end constraints. The theoretical results are complemented by numerical experiments.
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.
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.
Indirect methods for optimal control of hybrid PDE-dynamical / switching systems using relaxation
(2023)
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.
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.
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.
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.
Optimal boundary control of the isothermal semilinear Euler equation for gas dynamics on a network
(2023)
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.
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.
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.
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.
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.
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.
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.
The chapter reviews certain computational approaches to solve optimal control problems for evolution-type partial differential equations, where some control functions are limited to switching. The mechanism that enforces switching is modeled as integer restrictions. This brings a combinatorial aspect into the apart from switching already computationally very demanding optimization problems. Recently, great advances have been made to tackle such problems rigorously using relaxation and combinatorial integral approximation. An overview of these methods and the known theoretical results concerning convergence and error estimates are provided in a consistent manner. Further, we point to applications with benchmark character as well as to open problems.
We analyze a potentially risk-averse convex stochastic optimization problem, where the control is deterministic and the state is a Banach-valued essentially bounded random variable. We obtain strong forms of necessary and sufficient optimality conditions for problems subject to equality and conical constraints. We propose a Moreau–Yosida regularization for the conical constraint and show consistency of the optimality conditions for the regularized problem as the regularization parameter is taken to infinity.
A class of risk-neutral PDE-constrained generalized Nash equilibrium problems is introduced in which the feasible strategy set of each player is subject to a common linear elliptic partial differential equation with random inputs. In addition, each player’s actions are taken from a bounded, closed, and convex set on the individual strategies and a bound constraint on the common state variable. Existence of Nash equilibria and first-order optimality conditions are derived by exploiting higher integrability and regularity of the random field state variables and a specially tailored constraint qualification for GNEPs with the assumed structure. A relaxation scheme based on the Moreau-Yosida approximation of the bound constraint is proposed, which ultimately leads to numerical algorithms for the individual player problems as well as the GNEP as a whole. The relaxation scheme is related to probability constraints and the viability of the proposed numerical algorithms are demonstrated via several examples.