Finite convergence analysis and weak sharp solutions for variational inequalities

Abstract

Abstract. In this paper, we study the weak sharpness of the solution set of variational inequality problem (in short, VIP) and the finite convergence property of the sequence generated by some algorithm for finding the solutions of VIP. In particular, we give some characterizations of weak sharpness of the solution set of VIP without considering the primal or dual gap function. We establish an abstract result on the finite convergence property for a sequence generated by some iterative methods. We then apply such abstract result to discuss the finite termination property of the sequence generated by proximal point method, exact proximal point method and gradient projection method. We also give an estimate on the number of iterates by which the sequence converges to a solution of the VIP.

Finite convergence of extragradient-type methods for solving variational inequalities under weak sharp condition

“…Liu and Wu (2016a, b) further studied the weak sharp of solution set of VIs with respect to primal gap function. Recently, Al-Homidan et al (2016) used weak sharp solutions for the VIs without considering the primal or dual gap function to studied the finite termination property of sequences generated by iterative methods, such as the proximal point method, inexact proximal point method and gradient projection method. These results were also extended to non-smooth VIs as well as VIs on Hadamard manifolds and equilibrium problems (Al-Homidan et al 2017;Kolobov et al 2022Kolobov et al , 2021Nguyen et al 2020Nguyen et al , 2021.…”

On the finite termination of the gradient projection method

“…We investigate the finite convergence property of the sequence generated by the gradient projection method under the solution set is weakly sharp. The notion of weakly sharp occurs in optimization problems [4,15] and is an important tool in the analysis of the perturbation behavior of (1.1) as well as in the convergence analysis of algorithms designed to solve the problem; see [1,6,10,11,12,15] and the references therein. This notion was extended by Patriksson [14] to the variational inequality problem and developed by Marcotte and Zhu [10].…”

“…A remarkable feature of problem (1.1) is that the inverse strong monotonicity of the gradient ∇ f is satisfied automatically (see, e.g., [2,Theorem 18.5]). Moreover, Al-Homidan, Ansari and Nguyen [1] gave estimates on the number of iterates by which the sequence generated by the gradient projection method converges to a solution when S is weakly sharp and F is monotone, strongly pseudomonotone [1, Definition 1 (d)] and Lipschitz continuous. More precisely, they showed that the upper bound of the iteration number needed to achieve termination depends on the modulus of the strong pseudomonotonicity and the Lipschitz constant of F, the distance from the initial point x 0 to S, and the modulus of the weak sharpness of S [1, Theorem 6].…”

“…([1, Theorem 2]) If the solution set S of (1.1) is weakly sharp with modulus α > 0, then, for any x ∈ C, we havex − P S (x), ∇ f (x) ≥ αdist(x, S). (2.3)…”

Weak sharpness and finite convergence for mixed variational inequalities

“…Marcotte and Zhu [25] characterized weak sharpness of the solution set of a variational inequality in term of its dual gap function and studied finite convergence of sequences generated by some algorithms for solving variational inequalities under weak sharpness of the solution sets. Later, weak sharpness of solutions and its applications to the finite convergence property of methods for finding solutions of varitional inequalities have been investigated by many authors (see, e.g., [3,17,23,24,26,27,32,33,34] and references therein). Some authors extended and established the concept of weak sharp solutions to general variational inequalities, e.g., set-valued variational inequalities [2,31], variational-type inequalities [19], nonsmooth variational inequalities [4] and mixed variational inequalities [16].…”