Optimization and Variational Analysis

Optimization or, more generally, modern variational analysis, can be viewed as emerging from the calculus of variations and mathematical programming. One of the most characteristic features of this discipline is the intrinsic presence of nonsmoothnees, that is, the necessity to deal with nondifferentiable functions, sets with nonsmooth boundaries, and set-valued mappings. In fact, even the simplest problems in optimal control are intrinsically nonsmooth, in contrast to the classical calculus of variations. This is mainly due to pointwise constraints on control functions that often take only discrete values as in typical problems of automatic control.

Variational analysis and optimization cover many techniques coming from different mathematical fields such as: topology, convex analysis, integral and differential calculus, linear algebra, among others.

In the AM2V, our research is focused in the following topics:

1. Asymptotic behavior of trajectories that are solutions (in some sense) of: optimization problems, equations, differential inclusions.

2. Global and asymptotic properties of dynamical systems in discrete time coming from:

• algorithms for solving equations, optimization problems, variational inequalities.
• control problems related to the sustainable management of natural resources.

3. Topological, geometrical and algebraic sufficient conditions for set-valued mappings for obtaining existence, regularity and stability of some variational problems.

4. Characterization of several behaviors of functions and sets in term of the nonsmooth first order information (generalized derivatives, sudifferentials, tangent and normal cones).

Besides the theoretical areas listed above, our research interests touches some application fields as:

• Modeling, control and optimization for the management of forestry resources.
• Control and optimization of bioprocesses.
• Control and viability for the management of marine resources.