A second-order dynamical system for equilibrium problems

Abstract

We consider a second order dynamical system for solving equilibrium problems in Hilbert spaces. Under mild conditions, we prove existence and uniqueness of strong global solution of the proposed dynamical system. We establish the exponential convergence of trajectories under strong pseudo monotonicity and Lipschitz-type conditions. We then investigate a discrete version of the second order dynamical system, which leads to a fixed point type algorithm with inertial effect and relaxation. The linear convergence of this algorithm is established under suitable conditions on parameters. Finally some numerical experiments are reported confirming the theoretical results.

Relaxed-Inertial Proximal Point Algorithms for Nonconvex Equilibrium Problems with Applications

“…Even though the proximal operator is mostly considered in the literature with respect to the whole underlying space, in papers such as [7,17,18] it is taken on a certain set. The same algorithm has been proposed for solving (EP) in [24] (of which we became aware only when this paper was complete, hence the possible redundancies) as a relaxed inertial extragradient method and [48], via an explicit discretization of a second-order dynamical system, under different hypotheses that include the convexity of f in the second variable. Even if this is not explicitly acknowledged, in these works the proximal operator is considered with respect to the underlying set of (EP), too.…”

“…(ii) In virtue of relations (3.2) and (3.4), K should be assumed to be an affine subspace and not only as a closed and convex set. We note that the authors in [2,3,16,35] considered their algorithms on the whole space, while in [24,48] K was taken closed and convex, however the issue of staying feasible after the extrapolation step does not seem to have been taken into consideration.…”

“…In subsequent work, we plan to provide similar results achieved under a more general quasiconvexity assumption on the involved bifunction in its second variable. Moreover, we aim to investigate the extension to the quasiconvex framework considered in this work of the modified relaxed inertial extragradient method proposed in [24,48] in the convex case.…”

“…The main contribution of this work consists in showing that the relaxedinertial proximal point algorithm for solving equilibrium problems (RIPPA-EP from now on) introduced in [2,3] for solving monotone inclusion problems and extended to equilibrium problems in [24,48] in the convex case remains convergent, under slightly different, but still standard assumptions, when the employed bifunction is strongly quasiconvex in its second variable and pseudomonotone, following thus in a certain sense our recent works [18, 24,28,33]. Therefore, the influence of B.T.…”

Third Order Dynamical Systems for the Sum of Two Generalized Monotone Operators

“…This research direction has become increasingly attractive as it can provide new insights into optimization results and lead to interesting findings. Among the emerging research directions, there is a line of works that uses ordinary differential equations (ODEs) to design algorithms for optimization problems [2,3,9,14], variational inequalities [18,27,34,43], monotone inclusions [1,5,6], fixed point problems [15,17] and equilibrium problems [20,36,42,45]. Using ODE interpretation not only provides a better understanding of Nesterov's scheme, but also helps design new schemes with similar convergence rates.…”