View this talk here: https://www.youtube.com/watch?v=WIxhvk2vNQY

Annamaria Barbagallo and Serena Guarino Lo Bianco

The aim of the talk is to consider a general oligopolistic market equilibrium problem in which each firm produces several commodities. The equilibrium condition is expressed by a generalization of the Cournot-Nash principle and is characterized by a tensor variational inequality in which every operator is defined in a tensor Hilbert space. As a consequence, tensor variational inequalities, recently introduced in [1], have a fundamental role to study the model. For this reason, we show existence and regularity results for solutions to such inequalities. Moreover, we investigate on the convergence of solutions to regularized tensor variational inequalities to a solution to a tensor variational inequality making use of the set convergence in Kuratowski’s sense (see [2]). Then, we focus our attention on

some numerical schemes based on the projection opertor to compute the solutions (see [3]). Finally, we apply the theoretical results to our model.

References

[1] A. Barbagallo, S. Guarino Lo Bianco, Variational inequalities on a class of structured tensors, J. Nonlinear Conv. Anal. 19 (2018) 711 – 729.

[2] A. Barbagallo, S. Guarino Lo Bianco, On ill-posedness and stability of tensor variational

inequalities: application to an economic equilibrium, J. Global Optim. (2019)

https://doi.org/10.1007/s10898-019-00788-9.

[3] A. Barbagallo, S. Guarino Lo Bianco, G. Toraldo, Tensor variational inequalities: theoretical results, numerical methods and applications to an economic equilibrium model, J. Nonlinear Var. Anal. 4 (2020) 87 – 105.