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

  

Publications

  1. 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]
  2. 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]
  3. 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]
  4. Seeger A, Sossa D. Critical angles between two convex cones II. Special cases. TOP 2015. [Digital version] [Bibtex]
  5. Seeger A, Sossa D. Critical angles between two convex cones I. General theory. TOP 2015. [Digital version] [Bibtex]
  6. Seeger A, Sossa D. Complementarity problems with respect to Loewnerian cones. J. Global Optim. 2015;62(2):299–318. [Digital version] [Bibtex]
  7. Aguirre P. Bifurcations of two-dimensional global invariant manifolds near a non-central saddle-node homoclinic orbit. SIAM Journal on Applied Dynamical Systems 2015;14(3):1600-1644. [Digital version] [Bibtex]
  8. Piazza A, Pagnoncelli B. K. The stochastic Mitra-Wan forestry model: risk neutral and risk averse cases. Jornal of Economics 2015;115:175-194. [Digital version] [Bibtex]
  9. Piazza A, Roy S. Deforestation and optimal management. Journal of Economic Dynamics and Control 2015;53:15-27. [Digital version] [Bibtex]
  10. Colombo G, Dinh H. N, Henrion R, Mordukhovich B. Discrete Approximations of a Controlled Sweeping Process. Set-Valued and Var. Anal. 2015;23:69-86. [Digital version] [Bibtex]
  11. Dinh H. N, Nam N. M, Rector R. B. A Unified Approach to Convex and Convexified Generalized Differentiation of Nonsmooth Functions and Set-Valued Mappings. Vietnam Journal of Mathematics 2014;42(4):479-497. [Digital version] [Bibtex]
  12. Aguirre P. A general class of predation models with multiplicative Allee effect. Nonlinear Dynamics 2014;78:629-648. [Digital version] [Bibtex]
  13. Aguirre P. On two families of stochastic predation models with Allee effect. Scientia Series A: Mathematical Sciences 2014;25:96-107. [Digital version] [Bibtex]
  14. Piazza A, Pagnoncelli B. K. The optimal harvesting problem under price uncertainty. Annals of Operations Research 2014;217(1):425-445. [Digital version] [Bibtex]
  15. Aguirre P, Krauskopf B, Osinga H. M. Global invariant manifolds near a Shilnikov homoclinic bifurcation. Journal of Computational Dynamics 2014;1(1):1-38. [Digital version] [Bibtex]
  16. Attouch H, Peypouquet J, Redont P. A dynamical approach to an inertial forward-backward algorithm for convex minimization. SIAM Journal on Optimization 2014;24(1):232-256. [Digital version] [Bibtex]
  17. Briceño L. Forward-Douglas-Rachford splitting and forward- partial inverse method for solving monotone inclusions. Optimization 2014. [Digital version] [Bibtex]
  18. Gajardo P, Ramirez H, Rodriguez J. Tools for improving feeding strategies in a SBR with several species. Bioprocess Biosyst. Eng. 2014;37:63-70. [Digital version] [Bibtex]
  19. Gajardo P, Seeger A. Equilibrium problems involving the Lorentz cone. J. Glob. Optim. 2014;58:321-340. [Digital version] [Bibtex]
  20. Aguirre P, Flores J, González-Olivares E. Bifurcations and global dynamics in a predator-prey model with a strong Allee effect on prey, and a ratio-dependent functional response. Nonlinear Analysis: Real World Applications 2014;16:235-249. [Digital version] [Bibtex]
  21. Ramirez H, Seeger A, Sossa D. Commutation Principle for Variational Problems on Euclidean Jordan Algebras. SIAM J. Optim. 2013;23(2):687-694. [Digital version] [Bibtex]
  22. Noun N, Peypouquet J. Forward-backward penalty scheme for constrained convex minimization without inf-compactness. Journal of Optimization Theory and Applications 2013;158(3):787-795. [Digital version] [Bibtex]
  23. Aguirre P, Krauskopf B, Osinga H. M. Global invariant manifolds near homoclinic orbits to a real saddle: (non)orientability and flip bifurcation. SIAM Journal on Applied Dynamical Systems 2013;12(4):1803-1846. [Digital version] [Bibtex]
  24. Correa R, Gajardo P, Thibault L, Zagrodny D. Existence of minimizers on drops. SIAM J Optim 2013;23(2):1154-1166. [Digital version] [Bibtex]
  25. Bayen T, Gajardo P, Mairet F. Optimal Synthesis for the Minimum Time Control Problems of Fed-Batch Bioprocesses for Growth Functions with Two Maxima. J Optim Theory Appl 2013;158(2):521-553. [Digital version] [Bibtex]
  26. Gajardo P, Seeger A. Solving inverse cone-constrained eigenvalue problems. Numerische Mathematik 2013;123(2):309-331. [Digital version] [Bibtex]
  27. 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]
  28. 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]
  29. Colombo G, Henrion R, Dinh H. N, Mordukhovich B. Optimal control of the sweeping process. DCDIS Series B: Applications & Algorithms 2012;1-2b(19):117-159. [Digital version] [Bibtex]
  30. Frankel P, Peypouquet J. Lagrangian-penalization algorithm for constrained optimization and variational inequalities. Set-Valued and Variational analysis 2012;20(2):169-185. [Digital version] [Bibtex]
  31. Briceño L. A Douglas-Rachford splitting method for solving equilibrium problems. Nonlinear Anal. 2012;75:6053-6059. [Digital version] [Bibtex]
  32. Gajardo P, Seeger A. Reconstructing a matrix from a partial sampling of Pareto eigenvalues. Comput Optim Appl 2012;51(3):1119-1135. [Digital version] [Bibtex]
  33. Piazza A, Khan M. A. On the Mitra-Wan Forestry Model: A Unified Analysis. Journal of Economic Theory 2012;147(1):230-260. [Digital version] [Bibtex]
  34. Peypouquet J. Coupling the gradient method with a general exterior penalization scheme for convex minimization. Journal of Optimization Theory and Applications 2012;153(1):123-138. [Digital version] [Bibtex]
  35. Understanding the Shilnikov chaos via the computation of global invariant manifolds; 2011. [Digital version] [Bibtex]
  36. Aguirre P, Doedel E, Krauskopf B, Osinga H. Investigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields. Discrete and Continuous Dynamical Systems - Series A 2011;29(4):1309-1344. [Digital version] [Bibtex]
  37. Briceño L, Combettes P. L, Pesquet J. C, Pustelnik N. Proximal algorithms for multicomponent image processing. J. Math. Imaging Vision 2011;41:3-22. [Digital version] [Bibtex]
  38. Briceño L. Outer approximation method for constrained composite fixed point problems involving Lipschitz pseudo contractive operators. Numer. Funct. Anal. Optim. 2011;32:1099-1115. [Digital version] [Bibtex]
  39. Briceño L, Combettes P. L. A monotone+skew splitting model for composite monotone inclusions in duality. SIAM J. Optim. 2011;21:1230-1250. [Digital version] [Bibtex]
  40. Mairet F, Bernard O, Ras M, Lardon L, Steyer J. Modeling anaerobic digestion of microalgae using ADM1. Bioresource Technology 2011;(102):6823-6829. [Digital version] [Bibtex]
  41. Piazza A, Khan M. A. Optimal Cyclicity and Chaos in the 2-Sector RSS Model: An Anything-Goes Construction. Journal of Economic Behavior and Organization 2011;80(3):397-417. [Digital version] [Bibtex]
  42. Attouch H, Czarnecki M.-O, Peypouquet J. Coupling forward-backward with penalty schemes and parallel splitting for constrained variational inequalities. SIAM Journal on Optimization 2011;21(4):1251-1274. [Digital version] [Bibtex]
  43. Attouch H, Cabot A, Frankel P, Peypouquet J. Alternating proximal algorithms for constrained variational inequalities. Application to domain decomposition for PDE's. Nonlinear Analysis: Theory, Methods & Applications 2011;74(18):7455-7473. [Digital version] [Bibtex]
  44. Piazza A, Khan M. A. An Overview of Turnpike Theory: Towards the Discounted Deterministic Case. Advances on Mathematical Economics 2011;14:39-67. [Digital version] [Bibtex]
  45. Piazza A, Khan M. A. Classical Turnpike Theory and the Economics of Forestry. Journal of Economic Behavior and Organization 2011;79(3):194-210. [Digital version] [Bibtex]
  46. Gajardo P, Harmand J, Ramirez H, Rapaport A. Minimal time bioremediation of natural water resources. Automatica 2011;47(8):1764-1769. [Digital version] [Bibtex]
  47. Piazza A, Khan M. A. The Concavity Assumption on Felicities and Asymptotic Dynamics in the RSS model. Set Valued and Variational Analysis 2011;19(1):135-156. [Digital version] [Bibtex]
  48. Álvarez F, Peypouquet J. A unified approach to the asymptotic almost-equivalence of evolution systems without Lipschitz conditions. Nonlinear Analysis: Theory, Methods & Applications 2011;74(11):3440-3444. [Digital version] [Bibtex]
  49. Gajardo P, Peña-Torres J, Ramirez H. Harvesting economic models and catch-to-biomass dependence: the case of small pelagic fish. Natural Resource Modeling 2011;24(2):268-296. [Digital version] [Bibtex]
  50. De-Lara M, Gajardo P, Ramirez H. Viable states for monotone harvest models. Syst. Control Lett. 2011;60:192-197. [Digital version] [Bibtex]
  51. Attouch H, Czarnecki M.-O, Peypouquet J. Prox-Penalization and Splitting Methods for Constrained Variational Problems. SIAM Journal on Optimization 2011;21(1):149-173. [Digital version] [Bibtex]
  52. Piazza A, Khan M. A. The Economics of Forestry and a Set-Valued Turnpike of the Classical Type. Nonlinear Analysis 2011;74(1):171-181. [Digital version] [Bibtex]
  53. Attouch H, Briceño L, Combettes P. L. A parallel splitting method for coupled monotone inclusions. SIAM J. Control Optim. 2010;48:3246-3270. [Digital version] [Bibtex]
  54. Bravo M, Briceño L, Cominetti R, Cortés C, Martínez F. An integrated behavioral model of the land-use and transport systems with network congestion and location externalities. Transp. Res. Part B: Methodological 2010;44:584-596. [Digital version] [Bibtex]
  55. Piazza A, Khan M. A. On the Non-existence of Optimal Programs in the Robinson-Solow-Srinivasan (RSS) Model. Economic Letters 2010;109(2):94-98. [Digital version] [Bibtex]
  56. Khan M, Piazza A. About optimal harvesting policies for a multiple species forest without discounting. Journal of Economics 2010;100(3):217-233. [Digital version] [Bibtex]
  57. Álvarez F, Peypouquet J. Asymptotic almost-equivalence and ergodic convergence of Lipschitz evolution systems in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2010;73(9):3018-3033. [Digital version] [Bibtex]
  58. Peypouquet J, Sorin S. Evolution equations for maximal monotone operators: asymptotic analysis in continuous and discrete time. J. Convex Anal. 2010;17(3-4):1113—1163. [Digital version] [Bibtex]
  59. Khan M. A, Piazza A. On Uniform Convergence of Undiscounted Optimal Programs in the Mitra-Wan Forestry model: The Strictly Concave Case. International Journal of Economic Theory 2010;6:57-76. [Digital version] [Bibtex]
  60. Correa R, Gajardo P, Thibault L. Various Lipschitz like properties for functions and sets I: Directional derivative and tangential characterizations. SIAM J. Optim. 2010;20(4):1766-1785. [Digital version] [Bibtex]
  61. Aguirre P, Gonzalez-Olivares E, Sáez E. Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect. SIAM Journal on Applied Mathematics 2009;69(5):1244–1262. [Digital version] [Bibtex]
  62. Aguirre P, González-Olivares E, Sáez E. Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect. Nonlinear Analysis: Real World Applications 2009;10(3):1401–1416. [Digital version] [Bibtex]
  63. Briceño L, Combettes P. L. Convex variational formulation with smooth coupling for multicomponent signal decomposition and recovery. Numer. Math. Theory Methods Appl. 2009;2:485-508. [Digital version] [Bibtex]
  64. Peypouquet J. Asymptotic convergence to the optimal value of diagonal proximal iterations in convex minimization. J. Convex Anal. 2009;16(1):277-286. [Digital version] [Bibtex]
  65. Álvarez F, Peypouquet J. Asymptotic equivalence and Kobayashi-type estimates for nonautonomous monotone operators in Banach spaces. Discrete Contin. Dyn. Syst. 2009;25(4):1109-1128. [Digital version] [Bibtex]
  66. Piazza A. The optimal harvesting problem with a land market: a characterization of the asymptotic convergence. Economic Theory 2009;40(1):113-138. [Digital version] [Bibtex]
  67. Piazza A, Rapaport A. Optimal control of renewable resources with alternative use. Mathematical and Computer Modelling 2009;50(1-2):260-272. [Digital version] [Bibtex]
  68. Cominetti R, Piazza A. Asymptotic convergence of optimal harvesting policies for a multiple species forest. athematics of Operations Research 2009;34(3):576-593. [Digital version] [Bibtex]
  69. Correa R, Gajardo P, Thibault L. Links between directional derivatives through multidirectional mean value inequalities. Math. Programming Ser. B 2009;116(1-2):55-77. [Digital version] [Bibtex]
  70. Gajardo P, Mazenc F, Ramírez H. Competitive exclusion principle in a model of chemostat with delays. Dynamics of Continuous, Discrete and Impulsive Systems Ser. A: Math. Anal. 2009;16(4a):253-272. [Digital version] [Bibtex]
  71. Briceño L, Cominetti R, Cortés C, Martínez F. An integrated behavioral model of land use and transport system: a hyper-network equilibrium approach. Netw. Spat. Econ. 2008;8:201-224. [Digital version] [Bibtex]
  72. Cominetti R, Peypouquet J, Sorin S. Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization. J. Differential Equations 2008;245(12):3753-3763. [Digital version] [Bibtex]
  73. Gajardo P, Ramírez H, Rapaport A. Minimal time sequential batch reactors with bounded and impulse controls for one or more species. SIAM J. Control Optim. 2008;47(6):2827-2856. [Digital version] [Bibtex]
  74. Duc D. M, Dinh H. N, Nguyen L. H. Lagrange multipliers theorem and saddle point optimality criteria in mathematical programming. J. Math. Anal. Appl. 2006;323(1):441-455. [Digital version] [Bibtex]
  75. Gajardo P, Seeger A. Higher-order spectral analysis and weak asymptotic stability of convex processes. J. Math. Anal. Appl. 2006;318(1):155-174. [Digital version] [Bibtex]
  76. Correa R, Gajardo P, Thibault L. Subdifferential representation formula and subdifferential criteria for the behavior of nonsmooth functions. Nonlinear Anal. 2006;65(4):864-891. [Digital version] [Bibtex]
  77. Correa R, Gajardo P. Eigenvalues of Set-valued operators in Banach spaces. Set-Valued Anal. 2005;13(1):1-19. [Digital version] [Bibtex]
  78. Alvarez F, Correa R, Gajardo P. Inner estimation of the eigenvalue set and exponential series solutions to differential inclusions. J. Convex Anal. 2005;12(1):1-11. [Digital version] [Bibtex]
  79. Gajardo P, Seeger A. Epsilon-eigenvalues of multivalued operators. Set-Valued Anal. 2003;11(3):273-296. [Digital version] [Bibtex]
  

Projects

  1. Núcleo Científico Milenio ICM/FIC RC130003: Information and coordination in networks
  2. Conicyt REDES: AM2V-MODEMAT network on optimization and control
  3. FONDECYT 1140720: Deterministic and stochastic models in discrete time: some applications to exploitation of natural resources and general equilibrium theory.
  4. FONDECYT 1140829: Dynamical systems and algorithms for optimization and equilibrium problems involving nonsmooth and nonconvex functions
  5. ECOS C13E03: Dynamical systems and algorithms in nonsmooth and nonconvex optimization problems

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