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:
- Gajardo P, Seeger A. Solving inverse cone-constrained eigenvalue problems. Numerische Mathematik 2013;123(2):309-331. [Digital version] [Bibtex]
- Aguirre P, González-Olivares E, Torres S. Stochastic predator-prey model with Allee effect on prey. Nonlinear Analysis: Real World Applications 2013;14(1):768-779. [Digital version] [Bibtex]
- Briceño L, Combettes P. L. Monotone operator methods for Nash equilibria in non-potential games, in: Computational and Analytical Mathematics, D. Bailey, H. H. Bauschke, P. Borwein, F. Garvan, M. Théra, J. Vanderwerff, and H. Wolkowicz, Eds.,. 2013. [Digital version] [Bibtex]
- Peypouquet J, Frankel P. Lagrangian-penalization algorithm for constrained optimization and variational inequalities. Set-Valued and Variational analysis 2012;20(2):169-185. [Digital version] [Bibtex]
- Briceño L. A Douglas-Rachford splitting method for solving equilibrium problems. Nonlinear Anal. 2012;75:6053-6059. [Digital version] [Bibtex]
- FONDECYT Postdoctoral No. 3130497 - Global Dynamics and Bifurcations: Insight into Theory and Applications
- FONDECYT N° 1120239: On the regularity of Optimal Value Functions
- FONDECYT 3120054: Coupled equilibria in mixed-networks: modelization, methods, and applications
- FIC-Valparaiso: Herramientas cuantitativas para la recuperación sustentable de la merluza común
- STIC-AmSud OVIMINE: Optimization and Viability in Mining