Accelerated Methods for Saddle-Point Problem

Algorithms for solving variational inequalities and saddle point problems with some generalizations of Lipschitz property for operators

“…Proof. Similarly to [1], for any x 1 , x 2 ∈ Q x , let us estimate the following difference: (11), one can get…”

“…Then g(x) admits an inexact (δ, L, µ x )-model with δ = (D∆ + δ 0 ) and L defined in Remark 2. Applying k steps of the Fast Gradient method to the "outer" problem (10) and solving the "inner" problem (14) in linear time, we obtain an ε-solution (2) to the problem (1), where δ = O(ε). The total number of iterations does not exceed…”

“…Thus, after no more than k = 2 L µx log 2LR 2 ε iterations we obtain an ε-solution (2) of the considered saddle point problem (1).…”

“…In particular, in [1] authors considered the modification of some well-known scheme for speeding up methods for solving smooth saddle point problems. The smoothness, in such setting, means the Lipschitz continuity of all partial gradients of the objective function f (x, y).…”

“…Let us note, that due to the strong convexity-concavity of the considered saddle point problem, the convergence in argument takes place with the similar asymptotic behavior. Thus, we can consider the introduced definition of the ε-solution (2) for the strongly convex-concave saddle point problem (1).…”