# Operational Research

Operational Research studies analytical methods of problem-solving and decision-making models that are useful in the management of organizations. The research group consists of Gabriella Colajanni, Patrizia Daniele, Fabio Raciti, and Laura Scrimali.

**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. (Gabriella Colajanni, Patrizia Daniele, Fabio Raciti, 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, Fabio Raciti, 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, Fabio Raciti, 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. Moreover, Financial Optimization Models with short selling and transfer of securities are studied. (Gabriella Colajanni, Patrizia Daniele)

**Variational formulation of cybersecurity models:** a game theory approach to cybersecurity investment supply chain model with budget constraint is given. Moreover, models for cyber-security with non-linear constraints (through Lagrange multipliers), Projected Dynamic Systems and heuristics for the provisioning of 5G services are studied. (Gabriella Colajanni, 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. (Gabriella Colajanni, Patrizia Daniele, Laura Scrimali)

**Stochastic Optimization:** stochastic programming deals with the study of optimization problems in which the data are uncertain and are represented by random variables. Such models have numerous applications and describe situations in which decisions must be made before the uncertain flow of data can be observed. In particular, by means of two-stage stochastic optimization with recursion and two-stage variational inequalities, evacuation models in case of disastrous events and supply models of medical products in emergency situations are studied. Furthermore, by means of stochastic optimization and two- and three- stage variational inequalities, models of 5G service provision with UAVs in disaster scenarios are also studied. (Gabriella Colajanni, 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. Stackelberg models can be used to investigate the interactions between content creators and viewers in social media platforms (Laura Scrimali)

**Variational formulation of closed-loop supply chain:** equilibrium models applied to closed-loop supply chain for valuable collectibles are studied (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)

**Network Games:** in many economic and social models, agents (players) are thought of as nodes of a Network. The main feature of this class of games is that the utility function of each player depends on the strategies chosen by their neighbours in the graph of relationships. The research activity in this field focuses on the generalization of the constraints used, and in new applications. (Fabio Raciti)

**Optimization and inverse problems:** one of the most effective class of methods in the investigation of inverse problems arising from various fields of applied science, modeled with differential equations, yields to an optimization problem. The functional to be optimized depends on the solution of the direct problem. After a functional-analytic approximation, the numerical solution requires various techniques from free or constrained optimization. The applications considered focus on elasticity problems. (Fabio Raciti)

**Networks Optimization and supply chains:** a very large class of problems can be analyzed through network frameworks. Indeed, optimization models based on networks and supply chains are a very advantageous mathematical tool in order to provide guidance to decision-makers in the act of choice. Therefore, problems of various nature that apply to different real contexts and situations (from cloud computing services to financial markets with intermediation; from business management, including reverse supply chains in the context of recycling and e-commerce networks, to cybersecurity; from OR problems in Healthcare, such as the provisioning of reagents and swab tests during the Covid-19 Pandemic or the organ transplants with uncertain availability, to service requests management problems in a 5G network architecture with UAVs) are studied by means of (closed-loop) supply chain and network-based models, together with variational formulations, heuristic algorithms and exact methods. (Gabriella Colajanni, Patrizia Daniele)

**Teaching and learning of Operations Research using technologies:** paths for higher secondary schools’ students and teachers. The main aim is to improve students’ interest, motivation, and skills related to STEM disciplines by integrating Mathematics and Computer Science through Operations Research. We study teaching units and methods with examples and problems closely connected with students’ every-day life or with the industrial reality, balancing mathematical modeling and algorithmics. (Gabriella Colajanni)

**Integer Programming Problem:** mathematical optimization where some or all of the variables are restricted to be integers. Formulation of integer programming problems for contexts such as optimal management of water networks, optimal placement of virtual functions on UAVs for services based on 5G networks, cloud computing networks for IaaS providers, optimal programming formulation for university course timetabling (also curriculum-based). Heuristic and exact methods and algorithms. (Gabriella Colajanni, Patrizia Daniele)

##### Contacts

prof. Andrea Giacobbe