Stéphane Pia

Weighted variational inequalities in non-pivot Hilbert spaces with applications

We introduce variational inequalities defined in non-pivot Hilbert spaces and we show some existence results. Then, we prove regularity results for weighted variational inequalities in non-pivot Hilbert space. These results have been applied to the weighted traffic equilibrium problem. The continuity of the traffic equilibrium solution allows us to present a numerical method to solve the weighted variational inequality that expresses the problem. In particular, we extend the Solodov-Svaiter algorithm to the variational inequalities defined in finite-dimensional non-pivot Hilbert spaces. Then, by means of a interpolation, we construct the solution of the weighted variational inequality defined in a infinite-dimensional space. Moreover, we present a convergence analysis of the method.

Some classes of projected dynamical systems in Banach spaces and variational inequalities

We introduce some Projected Dynamical Systems based on metric and generalized Projection Operator in a strictly convex and smooth Banach Space. Then we prove that critical points of these systems coincide with the solution of a Variational Inequality.

Weighted Traffic Equilibrium Problem in Non Pivot Hilbert Spaces with Long Term Memory

In the paper we consider a weighted traffic equilibrium problem in a non‐pivot Hilbert space and prove the equivalence between a weighted Wardrop condition and a variational inequality with long term memory. As an application we show, using recent results of the Senseable Laboratory at MIT, how wireless devices can be used to optimize the traffic equilibrium problem.

Weighted Quasi-Variational Inequalities in Non-pivot Hilbert Spaces and Applications

The paper is devoted to the introduction of weighted quasi-variational inequalities in non-pivot Hilbert spaces. In the first part, we show some existence and regularity results for solutions to such weighted quasi-variational inequalities. The second part concerns the study of a new traffic equilibrium model, where weights and elastic demand occur. A weighted quasi-variational formulation for equilibrium conditions is provided. The general existence and regularity results obtained, in the first part, allow us to show the existence and the continuity of weighted elastic traffic equilibrium solutions. Finally, an example is provided.

A Panoramic View on Projected Dynamical Systems

The theory of generalized projections both in non-pivot Hilbert spaces and strictly convex and smooth Banach spaces is developed and the related theory of projected dynamical systems is highlighted. A particular emphasis is given to the equivalence between solutions of variational inequality and critical points of projected dynamical systems.

Duality for Weighted Traffic Equilibrium Problem and Computational Procedure

We consider the weighted traffic equilibrium problem introduced in [4] and we apply the infinite‐dimensional duality theorem developed in [2], obtaining the existence of Lagrange variables, which allow to describe the behavior of the weighted traffic. Moreover, making use of a regularity result we present a descritization method to compute the weighted traffic equilibrium solution.

Variational Inequalities for Time Dependent Financial Equilibrium with Price Constraints

We study a financial evolutionary problem, when variance-covariance matrices, sector financial holding volumes, instrument prices are time-dependent. As in P.Daniele [1], but assuming the realistic condition of a lower constraint for the price of each instrument, we give the evolutionary financial equilibrium condition, prove an equivalent variational inequality formulation and an existence result.

Projected Dynamical Inclusions in Reflexive Banach Spaces

The paper is devoted to study a class of projected dynamical inclusions in a reflexive Banach space introduced making use of a projection operator by E. Zarantonello in 1977 (see [9]). Then the equality of the set of critical points of these inclusions and the set of solutions to a variational inequality is shown.

Weighted Quasi‐Variational Inequalities in Transportation Networks

The aim of the paper is to present weighted quasi‐variational inequalities in non‐pivot Hilbert spaces that allow to introduce a new model for the transportation networks. Some existence and regularity results for solutions to weighted quasi‐variational inequalities are shown. These results are applied to the traffic network equilibrium model when data explicitly depend on time and travel demands are elastic, namely they have an effective dependence on the equilibrium solutions, and the traffic congestion is studied by means of weights.