The group conducts research in the fields of optimisation, convex and variational analysis, and dynamical systems. The use of convex analysis tools and techniques in relation to the analysis of variational problems is a well-established practice, which is widely exemplified in various application contexts: in this context, the notion of subdifferential for functions opens up the study of more general monotonic operators. In the last decade, convex analysis has also opened up to sub-Riemannian structures, such as Carnot groups, and has become a fundamental tool for the study of regularities of functions and solutions of differential equations.

The data defining a parametric optimisation problem are notoriously subject to measurement errors and approximation processes, which it may be inappropriate to overlook. This fact, together with the interest in the conditioning effect that, in general, data variations produce on the solutions of a problem, motivates the need to conduct stability and sensitivity analysis of optimisation problems (both scalar and vectorial), subject to various types of perturbation. Typically, this type of investigation focuses on the study of the behaviour of constraint systems, the set of solutions and the performance (or marginal) function associated with a class of problems, as the perturbation parameter varies. There are consolidated approaches to this type of question in the literature, essentially based on the use of stability properties for multifunctions, properties that are in various ways connected to appropriate concepts of regularity, the phenomenology of which is still being investigated. In these approaches, the quantitative analysis of conditions that propitiate the above-mentioned stability properties finds a convenient formal expression through elements of a calculus (normal, tangential and subdifferential) able to treat objects that, by their nature, escape an attractive differential representation, within which the geometry of convexity and its variational surrogates play a particularly important role.

Variational inequalities (VI) and, more generally, equilibrium problems (EP) provide a classical framework for representing problems from various fields. Many solution existence results for VI and EP are based on assumptions of monotonicity and/or convexity of the operator and bifunction involved. More generally, we consider variational inequalities (SVI) defined by multi-valued operators, and quasi-equilibrium problems (QEP), in which the constraint depends on the point and is therefore described by a multifunction. In the literature, the existence results of SVI solutions generally require assumptions of generalised monotonicity, in the Karamardian sense, of the operator involved; in this regard, the characterisation of the relative concept of maximality is of interest to study.

It is study the existence of solutions of SVI under the assumption of a different type of monotony, called Brezis (or topological) pseudomonotony of the operator, by approximating the starting operator with a succession of more regular operators.

In the case of QEPs, which are generally investigated with fixed point results for maps, different recently proposed techniques are used, which are based on regularised versions of the penalty method and which allow the existence of equilibria to be established under weak assumptions of coercivity, replacing the QEP problem with a succession of EP problems, and under weak assumptions of lower semicontinuity of the constraint map.

Further topics studied by the group concern the analysis of ordinary differential equations and partial derivative, as well as non-linear difference equations, with applications to game theory (oligopoly, congestion, evolutionary and market games) and other micro- and macro-economic domains (such as, for example, general economic equilibrium models, neo-Keynesian models and cyclical growth models), in addition to biological contexts, involving the coexistence of different species.

The above-mentioned contexts, all of which can be ascribed to the theory of dynamic systems, in discrete or continuous time, are analysed, mostly from a qualitative point of view, to study their equilibria, their stability, and the existence of periodic and chaotic orbits. In particular, the presence of complex dynamics can be demonstrated through methods of non-linear analysis based on topological grade theory and the notion of set covering relations for continuous functions, defined on subsets of the n-dimensional Euclidean space homeomorphic to the unit cube, that are contractive along some directions and expansive along the remainder. It should be noted that continuous-time dynamical systems described by differential equations with periodic coefficients can be studied through the same techniques used for discrete-time dynamical systems by analysing the Poincaré map associated with the system, whose fixed points correspond to the periodic solutions of the differential equation.

The study of the properties of the normal map and the notion of subdifferential on the Heisenberg group has had a considerable boost recently. The study of horizontally convex functions, of their horizontal subgradient map and, more generally, of horizontally monotone multifunctions is an important line of research: in recent years we have concentrated on the regularity problems of such multifunctions, on the properties of the sections of horizontally convex functions, and in particular on the possibility of extending classical Euclidean theory to the Monge-Ampère equations.

Participants:

• Andrea Giovanni Calogero
• Rita Pini
• Marina Pireddu
• Roberto Raimondo
• Amos Uderzo