RAVIEP
Recent Advances in Variational Inequalities and Equilibrium Problems I

Invited Session

Time Slot: Wednesday Morning

Room: 001

Chair: Patrizia Daniele

Variational inequalities, maximal elements and economic equilibrium problems

Time: 11:30

Maria Bernadette Donato (Department of Economics, University of Messina), Villanacci Antonio

Our main goal is to provide conditions under which the solutionsets to a maximal problem and to a generalized variational inequality problemcoincide and are nonempty. In choice theory, individual actors are assumed to have preferences on a set of alternatives, i.e., a binary relation on a choice set. Individuals choose a subset of the choice set using such preferences. The preference relation is often assumed to be transitive and complete. In the present talk, we consider the case of preferences without completeness and transitivity. In this context, the individual look for an element in the choice set which is not strictly preferred by any other element in the set, i.e., a maximal element. The way we use to find maximal elements is to follow a variational inequality approach. Finally, we apply the obtained results to get a proof of the existence of equilibria in general equilibrium models.

Disaster Management with UAVs and 5G technology: a Three-Stage Stochastic network-based Optimization Model

Time: 11:50

Daniele Sciacca (University of Catania), Colajanni Gabriella, Daniele Patrizia, Nagurney Anna

In a disaster situation, whether natural or man-made, slow-onset or sudden-onset, an efficient and efficacious management is paramount and can assist in the saving of lives, the reduction of pain and suffering, and the protection and restoration of infrastructure. 5G technology and Unmanned Aerial Vehicles (UAVs) have emerged as powerful tools for disaster management which allow service providers and decision-makers to restore connectivity and provide much needed services in disaster areas. In this talk, we propose a three-stage stochastic optimization model in which each of the three stages represents a phase of disaster management: the preparedness, the response, and the recovery/reconstruction phase. Furthermore, managers need to consider multiple disaster scenarios with different probabilities of occurrence so as to be able to minimize the potential losses. The optimization model is based on a multi-level network made up of users and devices on the ground, which require 5G services, the controller UAVs, which handle requests for services, and a fleet of UAVs, which execute the services. The proposed model allows providers to satisfy service requests (favoring those with higher priority) while minimizing the overall cost and by determining the optimal flows and the optimal management of additional resources. A variational formulation of the proposed optimization model, ensuring the existence of a solution, and a numerical example, validating the effectiveness of the proposed model, are provided.

Generalised Nash equilibrium seeking under partial information

Time: 12:10

Sergio Grammatico (Delft University of Technology)

We consider the generalised Nash equilibrium (GNE) problem under partial-decision information, i.e., where each player can only observe the decisions of its neighbours, while its cost function possibly depends on the actions of some other players. For this class of equilibrium problems, we study fully distributed, proximal-point and projected-pseudo-gradient algorithms. Finally, we discuss the numerical performance of our algorithms that can be accelerated via over-relaxation and inertial schemes.

Viscosity solutions in multi-leader-common-follower games

Time: 12:30

Jacqueline Morgan (Department of Economics and Statistics & CSEF, University of Naples Federico II), M.Beatrice Lignola

Two-stage games with multiple leaders having pessimistic behavior and one common follower are considered. These games may fail to have pessimistic solutions, even if the leader payoffs are linear and the optimal reaction of the follower to the leaders strategies is unique. In a suitable class of such games, approximate and viscosity solutions are introduced and are proven to exist under appropriate conditions, in line with the results obtained by the authors in J. Optim. Theory Appl., 173 (2017) for one-leader-one-follower games with a pessimistic leader, also called weak Stackelberg games or pessimistic bilevel optimization problems.