Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects

Attouch, Hedy; Maingé, Paul-Émile

ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 3, pp. 836-857.

In the setting of a real Hilbert space $\mathscr{H}$, we investigate the asymptotic behavior, as time t goes to infinity, of trajectories of second-order evolution equations            ü(t) + γ$\dot{u}$(t) + ∇ϕ(u(t)) + A(u(t)) = 0, where ∇ϕ is the gradient operator of a convex differentiable potential function ϕ : $\mathscr{H}\to ℝ$, A : $\mathscr{H}\to \mathscr{H}$ 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.

Classification : 34C35, 34D05, 65C25, 90C25, 90C30
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

[1] S. Adly, H. Attouch and A. Cabot, Finite time stabilization of nonlinear oscillators subject to dry friction - Nonsmooth mechanics and analysis. Adv. Mech. Math. 12 (2006) 289-304. | MR

[2] F. Alvarez, On the minimizing property of a second order dissipative system in Hilbert space. SIAM J. Control Optim. 38 (2000) 1102-1119. | MR | Zbl

[3] F. Alvarez, 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] F. Alvarez and H. Attouch, 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

[5] F. Alvarez and H. Attouch, An inertial proximal method for monotone operators via discretization of a nonlinear oscillator with damping. Set Valued Anal. 9 (2001) 3-11. | MR | Zbl

[6] F. Alvarez and H. Attouch, Convergence and asymptotic stabilization for some damped hyperbolic equations with non-isolated equilibria. ESAIM: COCV 6 (2001) 539-552. | Numdam | MR | Zbl

[7] F. Alvarez, H. Attouch, J. Bolte and P. Redont, 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

[8] A.S. Antipin, 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] H. Attouch and R. Cominetti, A dynamical approach to convex minimization coupling approximation with the steepest descent method. J. Diff. Equ. 128 (1996) 519-540. | MR | Zbl

[10] H. Attouch and M.-O. Czarnecki, Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria. J. Diff. Equ. 179 (2002) 278-310. | MR | Zbl

[11] H. Attouch and A. Soubeyran, Inertia and reactivity in decision making as cognitive variational inequalities. J. Convex. Anal. 13 (2006) 207-224. | MR | Zbl

[12] H. Attouch, D. Aze and R. Wets, 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

[13] H. Attouch, X. Goudou and P. Redont, 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

[14] H. Attouch, A. Cabot and P. Redont, 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

[15] H. Attouch, J. Bolte, P. Redont and A. Soubeyran, Alternating proximal algorithms for weakly coupled minimization problems. Applications to dynamical games and PDE's. J. Convex Anal. 15 (2008) 485-506. | MR | Zbl

[16] J.-B. Baillon and G. Haddad, Quelques propriétés des opérateurs angles-bornés et n-cycliquement monotones. Israel J. Math. 26 (1977) 137-150. | MR | Zbl

[17] J.-B. Baillon and A. Haraux, Comportement à l'infini pour les équations d'évolution avec forcing périodique. Arch. Rat. Mech. Anal. 67 (1977) 101-109. | MR | Zbl

[18] J. Bolte, Continuous gradient projection method in Hilbert spaces. J. Optim. Theory Appl. 119 (2003) 235-259. | MR | Zbl

[19] J. Bolte and M. Teboulle, Barrier operators and associated gradient-like dynamical systems for constrained minimization problems. SIAM J. Control Optim. 42 (2003) 1266-1292. | MR | Zbl

[20] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, Mathematical Studies. North-Holland (1973). | MR | Zbl

[21] A. Cabot, Inertial gradient-like dynamical system controlled by a stabilizing term. J. Optim. Theory Appl. 120 (2004) 275-303. | MR | Zbl

[22] T. Cazenave and A. Haraux, An introduction to semilinear evolution equations, Oxford Lecture Series in Mathematics and its Applications 13. Oxford University Press, Oxford (1998). | MR | Zbl

[23] P.L. Combettes and S.A. Hirstoaga, Visco-penalization of the sum of two operators. Nonlinear Anal. 69 (2008) 579-591. | MR | Zbl

[24] R. Cominetti, J. Peypouquet and S. Sorin, Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization. J. Diff. Equ. 245 (2008) 3753-3763. | MR | Zbl

[25] S. Ervedoza and E. Zuazua, Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl. 91 (2009) 20-48. | MR | Zbl

[26] S.D. Flam and J. Morgan, Newtonian mechanics and Nash play. Int. Game Theory Rev. 6 (2004) 181-194. | MR | Zbl

[27] I. Gallagher, Asymptotics of the solutions of hyperbolic equations with a skew-symmetric perturbation. J. Diff. Equ. 150 (1998) 363-384. | MR | Zbl

[28] J.K. Hale and G. Raugel, Convergence in gradient-like systems with applications to PDE. Z. Angew. Math. Phys. 43 (1992) 63-125. | MR | Zbl

[29] A. Haraux, Systèmes dynamiques dissipatifs et applications 17. Masson, RMA (1991). | MR | Zbl

[30] J. Hofbauer and S. Sorin, Best response dynamics for continuous zero-sum games. Discrete Continuous Dyn. Syst. Ser. B 6 (2006) 215-224. | MR | Zbl

[31] P.E. Maingé, Regularized and inertial algorithms for common fixed points of nonlinear operators. J. Math. Anal. Appl. 344 (2008) 876-887. | MR | Zbl

[32] D. Monderer and L.S. Shapley, Potential Games. Games Econ. Behav. 14 (1996) 124-143. | MR | Zbl

[33] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73 (1967) 591-597. | MR | Zbl

[34] B.T. Polyak, Introduction to Optimization. Optimization Software, New York (1987). | MR | Zbl

[35] R.T. Rockafellar, 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] M. Schatzman, A class of nonlinear differential equations of second order in time. Nonlinear Anal. 2 (1978) 355-373. | MR | Zbl