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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.

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.

Decomposition Methods for Time-Dependent Mixed-Integer Nonlinear Optimization Problems on Graphs
(2023)

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.

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.

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.

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.

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.

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 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.