## The 10 most recently published documents

Outer approximation for generalized convex mixed-integer nonlinear robust optimization problems
(2024)

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.

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.

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.

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.

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.

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.

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.

Implicit Peer Triplets in Gradient-Based Solution Algorithms for ODE Constrained Optimal Control
(2024)

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.

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.

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.