Variational Inequalities and equilibrium problems
Variational Inequalities and equilibrium problems
Contributed Session
Time Slot: Thursday Morning
Room: 001
Chair: Ricardo Almeida
Environmental damage reduction: when countries face conflicting objectives
Time: 11:30
M. Angeles Caraballo (Universidad de Sevilla), Zapata Asunción, Monroy Luisa, Mármol Amparo
Numerous international environmental agreements of countries have been aimedat limiting their polluting emissions. However, this is an arduous goal to achievesince it involves two often conflicting objectives for their governments: maximizingtheir monetary benefits and minimizing the perception of environmental damage,both of which depend on the level of pollution emitted by the set of all countries.Taking these two objectives into account, the situation is analyzed as a two-criteriagame where each country has a tolerance threshold with respect to global emissions.The approach considered makes it possible to deal with a key issue in the analysis:the fact that it is not possible to compare in monetary terms the results obtained whencountries act strategically in pursuit of their objectives. We show that depending onthe relationships between the thresholds for each country, different sets of equilibriaarise. A significant consequence of this research is that all of these equilibria providestrategies with a positive effect on emission reductions and can play an importantrole in reversing climate change.
Active Network and Price of Anarchy in Multi-Commodity Routing Games with Variable Demands
Time: 11:50
Valerio Dose (“Sapienza” – Università di Roma), Cominetti Roberto, Scarsini Marco
Equilibria in nonatomic routing games describe how traffic distributes on a network used by a large number of agents which are interested in minimizing their own delay. The mathematical model where such equilibria arise, is composed of a directed graph representing the network, and cost functions associated to each edge of the graph, which represent the delay experienced on an edge, as a positive nondecreasing function of the amount of traffic traveling on that same edge. In this framework, equilibria can be described as optimal solutions to a minimization problem. Nevertheless, they are usually not optimal with respect to a different, and more meaningful, objective function, which is the total delay experienced by all users of the network. For this reason, the Price of Anarchy has been considered in the literature, as an index of the inefficiency of equilibria. It is defined as the ratio of the total delay in an equilibrium and the optimal total delay.The traditional way of studying the Price of Anarchy has been determining upper bounds for it, which are tight in the worst case scenario. Recently, the Price of Anarchy has also been considered as a function of the traffic demand. In a previous publication, the authors analyze networks with a single commodity, which means that all users want to travel between the same origin and destination in the graph. In this situation, it is shown that with affine cost functions, the Price of Anarchy as a function of the demand, achieves local maxima only at points where it is not differentiable, and between two such points it can achieve at most one smooth local minimum. This confirms a behavior observed in previous empirical works on the subject. An important concept to be considered in this analysis is the one of active network, which is, for each level of demand, the set of edges which are used in equilibrium. If the active network is not constant in a neighborhood of some demand, we call such a demand a breakpoint. It is exactly at breakpoints that the Price of Anarchy can be non differentiable and achieve a local maximum.In the work we are presenting, we apply the same approach to networks with multiple commodities. In these type of networks, the users can be of different types, each of them traveling between different pairs of vertices of the graph. We consider the Price of Anarchy not only as a function of every possible vector of demands across all the commodities, but also as functions of a single real parameter where the vector of demands is given by a fixed demand function of the parameter. In both situations we can define the active network and breakpoints analogously to the single commodity case. We show that, with affine cost functions, if the demand function keeps the vector of demands on a line through the origin, then the behavior of the Price of Anarchy is the same as in the single commodity case. However, this is not true if the vector of demands moves on a line which does not go through the origin. In networks with continuously differentiable cost functions and strictly positive derivatives, we prove a result concerning the left and right derivative of the Price of Anarchy at breakpoints when the demands vary on a line through the origin. Finally, we also study how the active network varies in the whole spectrum of demands, focusing on the case of parallel networks where all types of users travel between the same two vertices, but each type uses a different set of paths.
Detection of the Origin of Movement: Graph-Theoretical Model and Data Processing
Time: 12:10
Marcello Sanguineti (University of Genova), Olga Matthiopoulou, Benoit Bardy, Antonio Camurri, Giorgio Gnecco, Denis Mottet
The agreement among participants in a survey is used to define a ground truth for the origin of full-body human movement, as it is perceived by observers. This may differ from the physical origin of movement, as it is defined by experts. Then, a computational model based on the theory of cooperative games is exploited to find automatically the perceived origin of movement. In more details, a transferable-utility game is built over a skeletal representation of the human body, based on a characteristic function related to how movement features change on adjacent vertices of the graph (joints), which are the players of the game. Then, the Shapley values of the joints are evaluated and used to extract a higher-level feature, which provides an estimate of the joint from which movement either originates or propagates. The method developed in [1,2] is refined (as in [3]) by considering a pre-processing of the available motion capture dataset, dealing with issues such as missing data, outlier detection, and noise reduction.This research received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 824160 EU (Project EnTimeMent) and from the Università Italo-Francese (project GALILEO 2021 no. G21\_89). G. Gnecco and M. Sanguineti are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilitàe le loro Applicazioni (GNAMPA—National Group for Mathematical Analysis, Probability and their Applications) of the Istituto Nazionale di Alta Matematica (INdAM—National Institute of Higher Mathematics).
[1] Kolykhalova, K., Gnecco, G, Sanguineti, M., Camurri, A., Volpe, G.: Graph-restricted game approach for investigating human movement qualities. Proc. of the 4th Int. Conf. on Movement Computing, 2017.
[2] Kolykhalova, K., Gnecco, G, Sanguineti, M., Camurri, A., Volpe, G.: Automated analysis of the origin of movement: An approach based on cooperative games on graphs. IEEE Trans. on Human Machine Systems, 2020.
[3] Matthiopoulou, O., Bardy, B, Gnecco, G., Mottet, D., Sanguineti, M., Camurri, A.: A computational method to automatically detect the perceived origin of full-body human movement and its propagation. Proc. of the 1st Int. Workshop on Multi-Scale on Movement Technologies, Utrecht, the Netherlands, 2020.
Optimization conditions for functionals dealing with variable order fractional derivatives and with dependence on an arbitrary function
Time: 12:30
Ricardo Almeida (University of Aveiro, Aveiro, Portugal)
In this work we combine two ideas: fractional deriv
atives of variable order and fractional derivatives depending on an another function. With such operators, we develop a variational problem theory by presenting necessary conditions of optimization for different kind of problems. An integration by parts formula is also proven, fundamental for the developing of our theory.