Proximal extrapolated gradient methods for variational inequalities

Abstract

The paper concerns with novel first-order methods for monotone variational inequalities. They use a very simple linesearch procedure that takes into account a local information of the operator. Also, the methods do not require Lipschitz continuity of the operator and the linesearch procedure uses only values of the operator. Moreover, when the operator is affine our linesearch becomes very simple, namely, it needs only simple vector–vector operations. For all our methods, we establish the ergodic convergence rate. In addition, we modify one of the proposed methods for the case of a composite minimization. Preliminary results from numerical experiments are quite promising.

A modified hybrid algorithm for solving a composite minimization problem in Banach spaces

“…Motivated and inspired by the results going on in this direction, the purpose of this paper is to use a modified hybrid algorithm for solving non-smooth composite minimization problem (1.1) in Banach spaces without the assumptions that f is Fréchet differentiable and ∇ f is L-Lipschitz continuous. Our results improve and extend the corresponding results in [7,8,11,12,16,17].…”

A First-Order Primal-Dual Algorithm with Linesearch

“…where x ∈ R n , y ∈ R m , A ∈ R m×n , and ∆ m , ∆ n denote the standard unit simplices in R m and R n , respectively. For this problem we study the performance of PDA, PDAL (Algorithm 1), Tseng's FBF method [17], and PEGM [15]. For comparison we use the primal-dual gap G(x, y), which can be easily computed for a feasible pair (x, y) as…”

“…Our proposed analysis of PDA exploits the idea of recent works [14,15] where several algorithms are proposed for solving a monotone variational inequality. Those algorithms are different from the PDA; however, they also use a similar extrapolation step.…”

Projected Subgradient Algorithms for Pseudomonotone Equilibrium Problems and Fixed Points of Pseudocontractive Operators

“…Equilibrium problems have been studied extensively in the literature (see, e.g., [2][3][4][5]). Many problems, such as variational inequalities [6][7][8][9][10][11][12][13][14][15], fixed point problems [16][17][18][19][20][21], and Nash equilibrium in noncooperative games theory [1,22], can be formulated in the form of Equation (3).…”