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Optimization and Control (math.OC)

Fri, 23 Jun 2023

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1.An Approximate Projection onto the Tangent Cone to the Variety of Third-Order Tensors of Bounded Tensor-Train Rank

Authors:Charlotte Vermeylen, Guillaume Olikier, Marc Van Barel

Abstract: An approximate projection onto the tangent cone to the variety of third-order tensors of bounded tensor-train rank is proposed and proven to satisfy a better angle condition than the one proposed by Kutschan (2019). Such an approximate projection enables, e.g., to compute gradient-related directions in the tangent cone, as required by algorithms aiming at minimizing a continuously differentiable function on the variety, a problem appearing notably in tensor completion. A numerical experiment is presented which indicates that, in practice, the angle condition satisfied by the proposed approximate projection is better than both the one satisfied by the approximate projection introduced by Kutschan and the proven theoretical bound.

2.Solving the Train Dispatching Problem in Large Networks by Column Generation

Authors:Maik Schälicke, Karl Nachtigall

Abstract: Disruptions in the operational flow of rail traffic can lead to conflicts between train movements, such that a scheduled timetable can no longer be realised. This is where dispatching is applied, existing conflicts are resolved and a dispatching timetable is provided. In the process, train paths are varied in their spatio-temporal course. This is called the train dispatching problem (TDP), which consists of selecting conflict-free train paths with minimum delay. Starting from a path-oriented formulation of the TDP, a binary linear decision model is introduced. For each possible train path, a binary decision variable indicates whether the train path is used by the request or not. Such a train path is constructed from a set of predefined path parts (\profiles{}) within a time-space network. Instead of modelling pairwise conflicts, stronger MIP formulation are achieved by a clique formulation. The combinatorics of speed profiles and different departure times results in a large number of possible train paths, so that the column generation method is used here. New train paths within the pricing-problem can be calculated using shortest path techniques. Here, the shadow prices of conflict cliques must be taken into account. When constructing a new train path, it must be determined whether this train path belongs to a clique or not. This problem is tackled by a MIP. The methodology is tested on practical size instances from a dispatching area in Germany. Numerical results show that the presented method achieves acceptable computation times with good solution quality while meeting the requirements for real-time dispatching.

3.Computational investigations of a two-class traffic flow model: mean-field and microscopic dynamics

Authors:Abderrahmane Habbal, Imad Kissami, Amal Machtalay, Ahmed Ratnani

Abstract: We address a multi-class traffic model, for which we computationally assess the ability of mean-field games (MFG) to yield approximate Nash equilibria for traffic flow games of intractable large finite-players. We introduce a two-class traffic framework, following and extending the single-class lines of \cite{huang_game-theoretic_2020}. We extend the numerical methodologies, with recourse to techniques such as HPC and regularization of LGMRES solvers. The developed apparatus allows us to perform simulations at significantly larger space and time discretization scales. For three generic scenarios of cars and trucks, and three cost functionals, we provide numerous numerical results related to the autonomous vehicles (AVs) traffic dynamics, which corroborate for the multi-class case the effectiveness of the approach emphasized in \cite{huang_game-theoretic_2020}. We additionally provide several original comparisons of macroscopic Nash mean-field speeds with their microscopic versions, allowing us to computationally validate the so-called $\epsilon-$Nash approximation, with a rate slightly better than theoretically expected.

4.Synchronous dynamic game on system observability considering one or two steps optimality

Authors:Yueyue Xu, Xiaoming Hu, Lin Wang

Abstract: This paper studies a system security problem in the context of observability based on a two-party non-cooperative asynchronous dynamic game. A system is assumed to be secure if it is not observable. Both the defender and the attacker have means to modify dimension of the unobservable subspace, which is set as the value function. Utilizing tools from geometric control, we construct the best response set under one-step or two-step optimality to minimize or maximize the value function. We find that the best response sets under one-step optimality are not single-valued maps, resulting in a variety of game outcomes. In the dynamic game considering two-step optimality, definition and existence conditions of lock and oscillation game modes are given. Finally, the best response under two-step optimality and the Stackelberg game equilibrium are compared.

5.Fast Approximation of Unbalanced Optimal Transport and Maximum Mean Discrepancies

Authors:Rajmadan Lakshmanan, Alois Pichler

Abstract: This contribution presents significant computational accelerations to prominent schemes, which enable the comparison of measures, even with varying masses. Concisely, we employ nonequispaced fast Fourier transform to accelerate the radial kernel convolution in unbalanced optimal transport approximation, building on the Sinkhorn algorithm. Accelerated schemes are presented as well for the maximum mean discrepancies involving kernels based on distances. By employing nonequispaced fast Fourier transform, our approaches significantly reduce the arithmetic operations to compute the distances from $\mathcal O(n^2)$ to $\mathcal O(n\log n)$, which enables access to large and high-dimensional data sets. Furthermore, we show some robust relation between the Wasserstein distance and maximum mean discrepancies. Numerical experiments using synthetic data and real datasets demonstrate the computational acceleration and numerical precision.

6.Optimal Sensor Placement with Adaptive Constraints for Nuclear Digital Twins

Authors:Niharika Karnik, Mohammad G. Abdo, Carlos E. Estrada Perez, Jun Soo Yoo, Joshua J. Cogliati, Richard S. Skifton, Pattrick Calderoni, Steven L. Brunton, Krithika Manohar

Abstract: Given harsh operating conditions and physical constraints in reactors, nuclear applications cannot afford to equip the physical asset with a large array of sensors. Therefore, it is crucial to carefully determine the placement of sensors within the given spatial limitations, enabling the reconstruction of reactor flow fields and the creation of nuclear digital twins. Various design considerations are imposed, such as predetermined sensor locations, restricted areas within the reactor, a fixed number of sensors allocated to a specific region, or sensors positioned at a designated distance from one another. We develop a data-driven technique that integrates constraints into an optimization procedure for sensor placement, aiming to minimize reconstruction errors. Our approach employs a greedy algorithm that can optimize sensor locations on a grid, adhering to user-defined constraints. We demonstrate the near optimality of our algorithm by computing all possible configurations for selecting a certain number of sensors for a randomly generated state space system. In this work, the algorithm is demonstrated on the Out-of-Pile Testing and Instrumentation Transient Water Irradiation System (OPTI-TWIST) prototype vessel, which is electrically heated to mimic the neutronics effect of the Transient Reactor Test facility (TREAT) at Idaho National Laboratory (INL). The resulting sensor-based reconstruction of temperature within the OPTI-TWIST minimizes error, provides probabilistic bounds for noise-induced uncertainty and will finally be used for communication between the digital twin and experimental facility.