Pseudomonotone Variational Inequalities: Convergence of the Auxiliary Problem Method

Generalized partially relaxed pseudomonotone variational inequalities and general auxiliary problem principle

“…In this paper, we intend based on a general auxiliary problem principle to present the approximation-solvability of a class of variational inequality problems (VIP) involving partially relaxed pseudomonotone mappings along with some modified results on Fréchet-differentiable functions that play a pivotal role in the development of a general framework for the auxiliary problem principle. Results thus obtained generalize/complement investigations of Argyros and 2 Auxiliary problem principle Verma [1], El Farouq [7], Verma [20], and others. For more details on general variational inequality problems and the auxiliary problem principle, we refer to .…”

“…Note that Corollary 3.6 is proved in [7,Theorem 4.1] with an additional imposition of the uniform continuity on the mapping T, but we feel that the uniform continuity is not required for the convergence purposes. …”

Interior Proximal Algorithm for Quasiconvex Programming Problems and Variational Inequalities with Linear Constraints

“…The following lemma has a crucial role in the well definedness of the sequence generated by Algorithm 2; it is a modification of Lemma 3.1 in [31]. We recall that λ min (A T A) is the minimum eigenvalue of the positive-definite symmetric matrix A T A.…”

Auxiliary problem principles for equilibria

“…If μ = 0, the well-known concepts of monotonicity, pseudomonotonicity and quasimonotonicity are recovered (see [14,18,31,32]). If μ > 0, the requirements are strengthened and the strong counterparts of the above monotonicity concepts defined: strong monotonicity has been often exploited in algorithmic frameworks (see [13]) while strong pseudomonotonicity has been considered mainly for variational inequalities (see [33,34]) and only very recently for more general EPs. [35] Similarly, if μ < 0, weaker concepts are introduced: weak monotonicity has been exploited in a few papers [36][37][38][39], while, to the best of our knowledge, weak quasimonotonicity has been used only in [29] to prove existence results for (EP).…”