In the setting of a real Hilbert space , we investigate the asymptotic behavior, as time t goes to infinity, of trajectories of second-order evolution equations ü(t) + γ(t) + ∇ϕ(u(t)) + A(u(t)) = 0, where ∇ϕ is the gradient operator of a convex differentiable potential function ϕ : , A : is a maximal monotone operator which is assumed to be λ-cocoercive, and γ > 0 is a damping parameter. Potential and non-potential effects are associated respectively to ∇ϕ and A. Under condition λγ2 > 1, it is proved that each trajectory asymptotically weakly converges to a zero of ∇ϕ + A. This condition, which only involves the non-potential operator and the damping parameter, is sharp and consistent with time rescaling. Passing from weak to strong convergence of the trajectories is obtained by introducing an asymptotically vanishing Tikhonov-like regularizing term. As special cases, we recover the asymptotic analysis of the heavy ball with friction dynamic attached to a convex potential, the second-order gradient-projection dynamic, and the second-order dynamic governed by the Yosida approximation of a general maximal monotone operator. The breadth and flexibility of the proposed framework is illustrated through applications in the areas of constrained optimization, dynamical approach to Nash equilibria for noncooperative games, and asymptotic stabilization in the case of a continuum of equilibria.
Mots clés : second-order evolution equations, asymptotic behavior, dissipative systems, maximal monotone operators, potential and non-potential operators, cocoercive operators, Tikhonov regularization, heavy ball with friction dynamical system, constrained optimization, coupled systems, dynamical games, Nash equilibria
@article{COCV_2011__17_3_836_0, author = {Attouch, Hedy and Maing\'e, Paul-\'Emile}, title = {Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {836--857}, publisher = {EDP-Sciences}, volume = {17}, number = {3}, year = {2011}, doi = {10.1051/cocv/2010024}, mrnumber = {2826982}, zbl = {1230.34051}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2010024/} }
TY - JOUR AU - Attouch, Hedy AU - Maingé, Paul-Émile TI - Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2011 SP - 836 EP - 857 VL - 17 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2010024/ DO - 10.1051/cocv/2010024 LA - en ID - COCV_2011__17_3_836_0 ER -
%0 Journal Article %A Attouch, Hedy %A Maingé, Paul-Émile %T Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects %J ESAIM: Control, Optimisation and Calculus of Variations %D 2011 %P 836-857 %V 17 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2010024/ %R 10.1051/cocv/2010024 %G en %F COCV_2011__17_3_836_0
Attouch, Hedy; Maingé, Paul-Émile. Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 3, pp. 836-857. doi : 10.1051/cocv/2010024. http://www.numdam.org/articles/10.1051/cocv/2010024/
[1] Finite time stabilization of nonlinear oscillators subject to dry friction - Nonsmooth mechanics and analysis. Adv. Mech. Math. 12 (2006) 289-304. | MR
, and ,[2] On the minimizing property of a second order dissipative system in Hilbert space. SIAM J. Control Optim. 38 (2000) 1102-1119. | MR | Zbl
,[3] Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space. SIAM J. Optim. 14 (2004) 773-782. | MR | Zbl
,[4] The heavy ball with friction dynamical system for convex constrained minimization problems, in Optimization, Namur (1998), Lecture Notes in Econom. Math. Systems 481, Springer, Berlin (2000) 25-35. | MR | Zbl
and ,[5] An inertial proximal method for monotone operators via discretization of a nonlinear oscillator with damping. Set Valued Anal. 9 (2001) 3-11. | MR | Zbl
and ,[6] Convergence and asymptotic stabilization for some damped hyperbolic equations with non-isolated equilibria. ESAIM: COCV 6 (2001) 539-552. | Numdam | MR | Zbl
and ,[7] A second-order gradient-like dissipative dynamical system with Hessian-driven damping. Application to optimization and mechanics. J. Math. Pures Appl. 81 (2002) 747-779. | MR | Zbl
, , and ,[8] Minimization of convex functions on convex sets by means of differential equations. Differ. Uravn. 30 (1994) 1475-1486 (in Russian). English translation: Diff. Equ. 30 (1994) 1365-1375. | MR | Zbl
,[9] A dynamical approach to convex minimization coupling approximation with the steepest descent method. J. Diff. Equ. 128 (1996) 519-540. | MR | Zbl
and ,[10] Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria. J. Diff. Equ. 179 (2002) 278-310. | MR | Zbl
and ,[11] Inertia and reactivity in decision making as cognitive variational inequalities. J. Convex. Anal. 13 (2006) 207-224. | MR | Zbl
and ,[12] Convergence of convex-concave saddle functions: Applications to convex programming and mechanics. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5 (1988) 537-572. | Numdam | MR | Zbl
, and ,[13] The heavy ball with friction method: The continuous dynamical system. Global exploration of local minima by asymptotic analysis of a dissipative dynamical system. Commun. Contemp. Math. 1 (2000) 1-34. | MR | Zbl
, and ,[14] The dynamics of elastic shocks via epigraphical regularization of a differential inclusion. Barrier and penalty approximations. Adv. Math. Sci. Appl. 12 (2002) 273-306. | MR | Zbl
, and ,[15] Alternating proximal algorithms for weakly coupled minimization problems. Applications to dynamical games and PDE's. J. Convex Anal. 15 (2008) 485-506. | MR | Zbl
, , and ,[16] Quelques propriétés des opérateurs angles-bornés et n-cycliquement monotones. Israel J. Math. 26 (1977) 137-150. | MR | Zbl
and ,[17] Comportement à l'infini pour les équations d'évolution avec forcing périodique. Arch. Rat. Mech. Anal. 67 (1977) 101-109. | MR | Zbl
and ,[18] Continuous gradient projection method in Hilbert spaces. J. Optim. Theory Appl. 119 (2003) 235-259. | MR | Zbl
,[19] Barrier operators and associated gradient-like dynamical systems for constrained minimization problems. SIAM J. Control Optim. 42 (2003) 1266-1292. | MR | Zbl
and ,[20] Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, Mathematical Studies. North-Holland (1973). | MR | Zbl
,[21] Inertial gradient-like dynamical system controlled by a stabilizing term. J. Optim. Theory Appl. 120 (2004) 275-303. | MR | Zbl
,[22] An introduction to semilinear evolution equations, Oxford Lecture Series in Mathematics and its Applications 13. Oxford University Press, Oxford (1998). | MR | Zbl
and ,[23] Visco-penalization of the sum of two operators. Nonlinear Anal. 69 (2008) 579-591. | MR | Zbl
and ,[24] Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization. J. Diff. Equ. 245 (2008) 3753-3763. | MR | Zbl
, and ,[25] Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl. 91 (2009) 20-48. | MR | Zbl
and ,[26] Newtonian mechanics and Nash play. Int. Game Theory Rev. 6 (2004) 181-194. | MR | Zbl
and ,[27] Asymptotics of the solutions of hyperbolic equations with a skew-symmetric perturbation. J. Diff. Equ. 150 (1998) 363-384. | MR | Zbl
,[28] Convergence in gradient-like systems with applications to PDE. Z. Angew. Math. Phys. 43 (1992) 63-125. | MR | Zbl
and ,[29] Systèmes dynamiques dissipatifs et applications 17. Masson, RMA (1991). | MR | Zbl
,[30] Best response dynamics for continuous zero-sum games. Discrete Continuous Dyn. Syst. Ser. B 6 (2006) 215-224. | MR | Zbl
and ,[31] Regularized and inertial algorithms for common fixed points of nonlinear operators. J. Math. Anal. Appl. 344 (2008) 876-887. | MR | Zbl
,[32] Potential Games. Games Econ. Behav. 14 (1996) 124-143. | MR | Zbl
and ,[33] Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73 (1967) 591-597. | MR | Zbl
,[34] Introduction to Optimization. Optimization Software, New York (1987). | MR | Zbl
,[35] Monotone operators associated with saddle-functions and mini-max problems, in Nonlinear operators and nonlinear equations of evolution in Banach spaces 2, 18th Proceedings of Symposia in Pure Mathematics, F.E. Browder Ed., American Mathematical Society (1976) 241-250. | MR | Zbl
,[36] A class of nonlinear differential equations of second order in time. Nonlinear Anal. 2 (1978) 355-373. | MR | Zbl
,Cité par Sources :