# Ricerca Operativa

Operational Research studies analytical methods of problem-solving and decision-making models that are useful in the management of organizations.

**Variational and quasi-variational inequalities:** a constrained optimization problem over a convex set can be characterized as a variational inequality. In the case of adaptive constraint set, it can be characterized as a quasi-variational inequality. Research in this field has led to existence results for both variational and quasi-variational inequalities in infinite dimensional spaces, endowed with weak or strong topology. The results are applied to equilibrium problems in transportation networks, financial networks and economic models, with time-dependent data, capacity constraints and solution-dependent equality constraints. (Patrizia Daniele, Laura Scrimali)

**Regularity of solutions to variational and quasi-variational inequalities:** the focus is on qualitative properties of solutions of variational and quasi-variational inequalities (continuity, Lipschitz condition, Hölder parametric stability, differentiability and norm estimate). (Patrizia Daniele, Laura Scrimali)

**Duality in infinite-dimensional spaces:** classic duality theory cannot be applied to problems in infinite dimensional spaces, since in most cases the set of constraints has empty interior. For this reason, duality is studied using separation theorem based on the concept of quasi-relative interior. (Patrizia Daniele)

**Solution methods for variational and quasi-variational inequalities:** computational procedures, such as Euler modified method, projection method, subgradient method, discretization, merit function, are studied. (Patrizia Daniele, Laura Scrimali)

**Equilibrium problems with dynamic and random data:** in the case of capacity constraints, dynamic flows or uncertain data, equilibria on traffic networks are characterized by means of generalized Wardrop conditions. In these situations, the models are represented by evolutionary or random variational inequalities. Random variational inequalities are also applied to the study of a spatial price network equilibrium model, in which consumers can weight both the transportation cost and the transportation time associated with the shipment of a given commodity. (Patrizia Daniele, Laura Scrimali)

**Financial model under risk and uncertainty:** when studying evolutionary financial models, the goal is maximizing assets and minimize risk and liabilities. Using duality theory, it is possible to obtain the optimal composition of assets and liabilities. (Patrizia Daniele)

**Variational formulation of cybersecurity models:** a game theory approach to cybersecurity investment supply chain model with budget constraint is given. (Patrizia Daniele)

**Lagrangian theory:** given a constrained optimization problem, it is possible to apply Lagrange duality theory and study the role of Lagrange multipliers. The meaning of multipliers allows one to understand better the model behavior. (Patrizia Daniele, Laura Scrimali)

**Bilevel optimization:** bilevel optimization problems can be applied to the pollution emission price problem. For this problem, the existence of Lagrange multipliers and optimality conditions are proved. (Laura Scrimali)

**Retarded equilibria:** retarded models can be formulated in terms of variational and quasi-variational inequalities. A Walras economic model is proposed with an integral memory term that represents previous equilibrium states. (Laura Scrimali)

**Inverse variational inequality:** this class of variational inequalities are used to describe control problems, such as control policies to regulate production and consumption of goods in spatial price network equilibrium models. (Laura Scrimali)

**Nash equilibrium:** several competition models on networks are described by the concept of Nash equilibrium. These models, such as telecommunication networks or information security networks, can be formulated as variational or quasi-variational inequalities. In addition, mixed equilibrium models are studied, in which some users follow the Nash equilibrium or the Wardrop principle, according to the market power. (Patrizia Daniele, Laura Scrimali)