On the Minimizing Property of a Second Order Dissipative System in Hilbert Spaces

Álvarez, Felipe

doi:10.1137/s0363012998335802

Cited by 267 publications

(298 citation statements)

References 11 publications

(12 reference statements)

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“…The following proposition will play an essential role in the convergence analysis (see also [1][2][3]16]). …”

Section: Lemma 6 Let X ∈ S and Set P := −∇ F (X)mentioning

confidence: 99%

“…Algorithms of inertial type result from the time discretization of differential inclusions of second order type (see [1,3]) and were first investigated in the context of the minimization of a differentiable function by Polyak [36] and Bertsekas [12]. The resulting iterative schemes share the feature that the next iterate is defined by means of the last two iterates, a fact which induces the inertial effect in the algorithm.…”

Section: Introductionmentioning

confidence: 99%

“…The resulting iterative schemes share the feature that the next iterate is defined by means of the last two iterates, a fact which induces the inertial effect in the algorithm. Since the works [1,3], one can notice an increasing number of research efforts dedicated to algorithms of inertial type (see [1][2][3]9,[16][17][18][19][24][25][26][27][28][30][31][32]34]). …”

Section: Introductionmentioning

confidence: 99%

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Gradient-type penalty method with inertial effects for solving constrained convex optimization problems with smooth data

2017

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We consider the problem of minimizing a smooth convex objective function subject to the set of minima of another differentiable convex function. In order to solve this problem, we propose an algorithm which combines the gradient method with a penalization technique. Moreover, we insert in our algorithm an inertial term, which is able to take advantage of the history of the iterates. We show weak convergence of the generated sequence of iterates to an optimal solution of the optimization problem, provided a condition expressed via the Fenchel conjugate of the constraint function is fulfilled. We also prove convergence for the objective function values to the optimal objective value. The convergence analysis carried out in this paper relies on the celebrated Opial Lemma and generalized Fejér monotonicity techniques. We illustrate the functionality of the method via a numerical experiment addressing image classification via support vector machines.

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“…The following proposition will play an essential role in the convergence analysis (see also [1][2][3]16]). …”

Section: Lemma 6 Let X ∈ S and Set P := −∇ F (X)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Gradient-type penalty method with inertial effects for solving constrained convex optimization problems with smooth data

2017

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“…The similarity between second-order Hamiltonian systems and the corresponding first-order gradient flows is wellknown in mechanical control systems [19], in dynamic optimization [20], [21], [22], and also in transient stability studies [23], [24], [25], but we are not aware of any result as general as Theorem III.1. In [23], [24], [25], statements 1) and 2) are proved under the more stringent assumptions that H λ has a finite number of isolated and hyperbolic equilibria.…”

Section: Parameterized Hamiltonian and Gradient-like Dynamics Anmentioning

confidence: 99%

“…Additionally, if H(x) constitutes an energy function and if a one-parameter transversality condition is satisfied, then the separatrices of system (6) can also be characterized accurately [23], [24], [26]. Also statement 3) can be refined under further structural assumptions on the potential function H(x), and various other minimizing properties can be deduced from the dynamics (6), see [19], [20], [21], [22].…”

Section: Parameterized Hamiltonian and Gradient-like Dynamics Anmentioning

confidence: 99%

Topological equivalence of a structure-preserving power network model and a non-uniform Kuramoto model of coupled oscillators

Dörfler

Bullo

2011

IEEE Conference on Decision and Control and European Control Conference

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Abstract-We study synchronization in the classic structurepreserving power network model proposed by Bergen and Hill. We find that, locally near the synchronization manifold, the phase and frequency dynamics of the power network model are topologically conjugate to the phase dynamics of a non-uniform Kuramoto model and decoupled exponentially stable dynamics for the frequencies. This topological conjugacy implies the equivalence of local exponential synchronization in power networks and in non-uniform Kuramoto oscillators. Hence, we can harness the results available for Kuramoto oscillators to analyze synchronization in power networks. We establish necessary and sufficient conditions for phase synchronization, sufficient conditions for frequency synchronization, and necessary and sufficient conditions for frequency synchronization with a uniform topology. These conditions also extend the results known for the classic first-order Kuramoto model and the second-order linear consensus protocols. Our synchronization conditions all share a common physical interpretation: the ratio of power inputs and dissipation has to be sufficiently uniform and the coupling in the network has to be sufficiently strong.

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An inertial primal‐dual fixed point algorithm for composite optimization problems

Wen¹,

Tang²,

Peng³

2022

Math Methods in App Sciences

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In this paper, we consider an inertial primal-dual fixed point algorithm (IPDFP) to compute the minimizations of the sum of a non-smooth convex function and a finite family of composite non-smooth convex functions, each one of which is composed of a non-smooth convex function and a bounded linear operator. This is a full splitting approach, in the sense that non-smooth functions are processed individually via their proximity operators. The convergence of the IPDFP is obtained by reformulating the problem to the sum of three non-smooth convex functions. Furthermore, we propose a preconditioning technique for the IPDFP. The key idea of the preconditioning technique is that the constant iterative parameters are updated self-adaptively in the iteration process. What's more, we also give a simple and easy way to choose the diagonal preconditioners while the convergence of the iterative algorithms is maintained. This work brings together and notably extends several classical splitting schemes, like the primal-dual method proposed by Chambolle and Pock, and the recent proximity algorithms of Charles et al. designed for the L 1 /TV image denoising model. The iterative algorithm is used for solving non-differentiable convex optimization problems arising in image processing.

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On the Minimizing Property of a Second Order Dissipative System in Hilbert Spaces

Cited by 267 publications

References 11 publications

Gradient-type penalty method with inertial effects for solving constrained convex optimization problems with smooth data

Gradient-type penalty method with inertial effects for solving constrained convex optimization problems with smooth data

Topological equivalence of a structure-preserving power network model and a non-uniform Kuramoto model of coupled oscillators

An inertial primal‐dual fixed point algorithm for composite optimization problems

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