COMPUTATION OF AVOIDANCE REGIONS FOR DRIVER ASSISTANCE SYSTEMS BY USING A HAMILTON-JACOBI APPROACH

International audienceWe consider the problem of computing safety regions, modeled as nonconvex backward reachable sets, for a nonlinear car collision avoidance model with time-dependent obstacles. The Hamilton-Jacobi-Bellman framework is used. A new formulation of level set functions for obstacle avoidance is given and sufficient conditions for granting the obstacle avoidance on the whole time interval are obtained, even though the conditions are checked only at discrete times. Different scenarios including various road configurations, different geometry of vehicle and obstacles, as well as fixed or moving obstacles, are then studied and computed. Computations involve solving nonlinear partial differential equations of up to five space dimensions plus time with nonsmooth obstacle representations, and an efficient solver is used to this end. A comparison with a direct optimal control approach is also done for one of the examples

Batch and Continuous Flow Adsorption of Phenolic Compounds from Olive Mill Wastewater: A Comparison between Nonionic and Ion Exchange Resins

The goals of this work were (i) to compare two anion ion exchange resins (IRA958 Cl and IRA67) and a non-ionic resin (XAD16) in terms of phenolic compounds adsorption capacity from olive mill wastewater, and (ii) to compare the adsorption capacity of the best resin on columns of different length. The ion exchange resins proved less performant than non-ionic XAD16 in terms of resin utilization efficiency (20% versus 43%) and phenolic compounds/COD enrichment factor (1.0 versus 2.5). The addition of volatile fatty acids did not hinder phenolic compounds adsorption on either resin, suggesting a non-competitive adsorption mechanism. A pH increase from 4.9 to 7.2 did not affect the result of this comparison. For the best performing resin (XAD16), an increase in column length from 0.5 to 1.8 m determined an increase in resin utilization efficiency (from 12% to 43%), resin productivity (from 3.4 to 7.6 g sorbed phenolics/kg resin) and phenolics/COD enrichment factor (from 1.2 to 2.5). An axial dispersion model with non-equilibrium adsorption accurately interpreted the phenolic compounds and COD experimental curves

Solving Constrained Piecewise Linear Optimization Problems by Exploiting the Abs-linear Approach

In dieser Arbeit wird ein Algorithmus zur Lösung von endlichdimensionalen Optimierungsproblemen mit stückweise linearer Zielfunktion und stückweise linearen Nebenbedingungen vorgestellt. Dabei wird angenommen, dass die Funktionen in der sogenannten Abs-Linear Form, einer Matrix-Vektor-Darstellung, vorliegen. Mit Hilfe dieser Form lässt sich der Urbildraum in Polyeder zerlegen, so dass die Nichtglattheiten der stückweise linearen Funktionen mit den Kanten der Polyeder zusammenfallen können. Für die Klasse der abs-linearen Funktionen werden sowohl für den unbeschränkten als auch für den beschränkten Fall notwendige und hinreichende Optimalitätsbedingungen bewiesen, die in polynomialer Zeit verifiziert werden können. Für unbeschränkte stückweise lineare Optimierungsprobleme haben Andrea Walther und Andreas Griewank bereits 2019 mit der Active Signature Method (ASM) einen Lösungsalgorithmus vorgestellt. Aufbauend auf dieser Methode und in Kombination mit der Idee der aktiven Mengen Strategie zur Behandlung von Ungleichungsnebenbedingungen entsteht ein neuer Algorithmus mit dem Namen Constrained Active Signature Method (CASM) für beschränkte Probleme. Beide Algorithmen nutzen die stückweise lineare Struktur der Funktionen explizit aus, indem sie die Abs-Linear Form verwenden. Teil der Analyse der Algorithmen ist der Nachweis der endlichen Konvergenz zu lokalen Minima der jeweiligen Probleme sowie die Betrachtung effizienter Berechnung von Lösungen der in jeder Iteration der Algorithmen auftretenden Sattelpunktsysteme. Die numerische Performanz von CASM wird anhand verschiedener Beispiele demonstriert. Dazu gehören akademische Probleme, einschließlich bi-level und lineare Komplementaritätsprobleme, sowie Anwendungsprobleme aus der Gasnetzwerkoptimierung und dem Einzelhandel.This thesis presents an algorithm for solving finite-dimensional optimization problems with a piecewise linear objective function and piecewise linear constraints. For this purpose, it is assumed that the functions are in the so-called Abs-Linear Form, a matrix-vector representation. Using this form, the domain space can be decomposed into polyhedra, so that the nonsmoothness of the piecewise linear functions can coincide with the edges of the polyhedra. For the class of abs-linear functions, necessary and sufficient optimality conditions that can be verified in polynomial time are given for both the unconstrained and the constrained case. For unconstrained piecewise linear optimization problems, Andrea Walther and Andreas Griewank already presented a solution algorithm called the Active Signature Method (ASM) in 2019. Building on this method and combining it with the idea of the Active Set Method to handle inequality constraints, a new algorithm called the Constrained Active Signature Method (CASM) for constrained problems emerges. Both algorithms explicitly exploit the piecewise linear structure of the functions by using the Abs-Linear Form. Part of the analysis of the algorithms is to show finite convergence to local minima of the respective problems as well as an efficient solution of the saddle point systems occurring in each iteration of the algorithms. The numerical performance of CASM is illustrated by several examples. The test problems cover academic problems, including bi-level and linear complementarity problems, as well as application problems from gas network optimization and inventory problems