# 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:

## Members

- Pablo Aguirre
- Luis Briceño
- Pedro Gajardo
- Hoang Nguyen Dinh
- Juan Peypouquet
- Adriana Piazza
- Matthieu Sebbah
- David Sossa

## Publications

- Colombo G, Henrion R, Dinh H. N, Mordukhovich B. Optimal control of the sweeping process over polyhedral controlled sets.
*Journal of Differential Equations*2016. [Digital version] [Bibtex] - Briceño L, Dinh H. N, Peypouquet J. Existence, stability and optimality for optimal control problems governed by maximal monotone operators.
*Journal of Differential Equations*2016;260(1):733-757. [Digital version] [Bibtex] - Frankel P, Garrigos G, Peypouquet J. Splitting methods with variable metric for KL functions and general convergence rates.
*Journal of Optimization Theory & Applications*2015;165:874-900. [Digital version] [Bibtex] - Seeger A, Sossa D. Critical angles between two convex cones II. Special cases.
*TOP*2015. [Digital version] [Bibtex] - Seeger A, Sossa D. Critical angles between two convex cones I. General theory.
*TOP*2015. [Digital version] [Bibtex]

## Projects

- Núcleo Científico Milenio ICM/FIC RC130003: Information and coordination in networks
- Conicyt REDES: AM2V-MODEMAT network on optimization and control
- FONDECYT 1140720: Deterministic and stochastic models in discrete time: some applications to exploitation of natural resources and general equilibrium theory.
- FONDECYT 1140829: Dynamical systems and algorithms for optimization and equilibrium problems involving nonsmooth and nonconvex functions
- ECOS C13E03: Dynamical systems and algorithms in nonsmooth and nonconvex optimization problems
- FONDECYT (postdoctorado) 3140060: Optimal control of nonconvex differential inclusions and applications
- FONDECYT Postdoctoral No. 3130497 - Global Dynamics and Bifurcations: Insight into Theory and Applications
- Núcleo Científio Milenio ICM/FIC P10-024F: Information and coordination in networks
- 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
- CONICYT Anillo ACT 88: Mathematical modeling for Industrial and Management Science Applications
- DYMECOS (Chilean associate team INRIA to EPI-MERE): Modeling, analysis and simulation of microbial ecosystems and natural resources
- SticAmsud MOMARE: Modelamiento Matematico para el Manejo de Recursos Naturales
- Fondecyt Nro. 11090254: Discrete time dynamical systems in the theory of Optimal Economic Growth, Environmental Economics and Management of Renewable Resources
- FONDECYT N 11090023: Asymptotic analysis of perturbed dynamical systems and applications
- ECOS-CONICYT C07E03: Viable control of discrete time systems and applications
- Fondecyt N 1080173: Characterization of various Lipschitz like properties of functions, sets, and set-valued mappings
- INRIA-CONICYT N 0701: Dynamical Modeling of Microbial Ecosystems ECOLOMICRO II
- SticAmsud MIFIMA: Mathematics, Informatics and Fisheries Management
- INRIA-CONICYT N 0601: Dynamical Modeling of Microbial Ecosystems
- Postdoctorado Fondecyt N 3060068: Mathematical modeling for the renewable resources management and nondifferentiable analysis
- ECOS-CONICYT C04E03: Characterization of nonsmooth functions and dynamical methods in optimization
- FONDEF D10I 1002: Tecnologia avanzada para ciudades del futuro
- MathAmSud 13MATH01: Variational Analysis and its applications to optimization, monotone operators, convex analysis, and Nash equilibrium problems