2,249 research outputs found
Stochastic mirror descent dynamics and their convergence in monotone variational inequalities
We examine a class of stochastic mirror descent dynamics in the context of
monotone variational inequalities (including Nash equilibrium and saddle-point
problems). The dynamics under study are formulated as a stochastic differential
equation driven by a (single-valued) monotone operator and perturbed by a
Brownian motion. The system's controllable parameters are two variable weight
sequences that respectively pre- and post-multiply the driver of the process.
By carefully tuning these parameters, we obtain global convergence in the
ergodic sense, and we estimate the average rate of convergence of the process.
We also establish a large deviations principle showing that individual
trajectories exhibit exponential concentration around this average.Comment: 23 pages; updated proofs in Section 3 and Section
Fixed-Time Stable Proximal Dynamical System for Solving MVIPs
In this paper, a novel modified proximal dynamical system is proposed to
compute the solution of a mixed variational inequality problem (MVIP) within a
fixed time, where the time of convergence is finite, and is uniformly bounded
for all initial conditions. Under the assumptions of strong monotonicity and
Lipschitz continuity, it is shown that a solution of the modified proximal
dynamical system exists, is uniquely determined and converges to the unique
solution of the associated MVIP within a fixed time. As a special case for
solving variational inequality problems, the modified proximal dynamical system
reduces to a fixed-time stable projected dynamical system. Furthermore, the
fixed-time stability of the modified projected dynamical system continues to
hold, even if the assumption of strong monotonicity is relaxed to that of
strong pseudomonotonicity. Connections to convex optimization problems are
discussed, and commonly studied dynamical systems in the continuous-time
optimization literature follow as special limiting cases of the modified
proximal dynamical system proposed in this paper. Finally, it is shown that the
solution obtained using the forward-Euler discretization of the proposed
modified proximal dynamical system converges to an arbitrarily small
neighborhood of the solution of the associated MVIP within a fixed number of
time steps, independent of the initial conditions. Two numerical examples are
presented to substantiate the theoretical convergence guarantees.Comment: 12 pages, 5 figure
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