Visco-penalization of the sum of two monotone operators

Combettes, Patrick L.; Hirstoaga, Sever Adrian

doi:10.1016/j.na.2007.06.003

Cited by 7 publications

(5 citation statements)

References 22 publications

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“…In the final section, we briefly outline situations where our results can be applied. Indeed, the convergence results obtained in Theorems 2.1 and 3.1 offer promising views on numerical optimization and algorithms (by time discretization as in [2], [4], [12], [22]), on the modeling of inertial dynamical approach to Nash equilibria in decision sciences and game theory (see [15], [23]), and on dissipative dynamical systems and PDE. 's (as in [6] for the damped wave equation).…”

Section: Introductionmentioning

confidence: 99%

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

Attouch¹,

Maingé²

2010

ESAIM: COCV

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Abstract.In the setting of a real Hilbert space H, we investigate the asymptotic behavior, as time t goes to infinity, of trajectories of second-order evolution equationswhere ∇φ is the gradient operator of a convex differentiable potential function φ : H → R, A : H → 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 λγ 2 > 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.Mathematics Subject Classification. 34C35, 34D05, 65C25, 90C25, 90C30.

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Section: Introductionmentioning

confidence: 99%

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

Attouch¹,

Maingé²

2010

ESAIM: COCV

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“…While this was answered in some form in [1] when A and B are normal cone operators, no result was offered in [2]. We conclude this paper by providing a complete and satisfying answer, relying on tools by Combettes and Hirstoaga [9] and [10], packed also into [6,Theorem 23.44].…”

Section: The Arag óN Artacho-campoy Algorithm (Aaca)mentioning

confidence: 94%

“…and we also assume that A + B is maximally monotone (10) which will make all results more tidy. (See also Remark 3.3 below.)…”

Section: The Aragón Artacho-campoy Algorithm (Aaca)mentioning

confidence: 99%

On the asymptotic behaviour of the Aragón Artacho–Campoy algorithm

Alwadani

Bauschke

Moursi

et al. 2018

Operations Research Letters

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Aragón Artacho and Campoy recently proposed a new method for computing the projection onto the intersection of two closed convex sets in Hilbert space; moreover, they proposed in 2018 a generalization from normal cone operators to maximally monotone operators. In this paper, we complete this analysis by demonstrating that the underlying curve converges to the nearest zero of the sum of the two operators. We also provide a new interpretation of the underlying operators in terms of the resolvent and the proximal average.

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“…Also various types of variational inclusions problems have been extended and generalized. For related work, please see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

Section: Definitionmentioning

confidence: 99%