General regularized nonconvex mixed equilibrium problems: iterative methods by auxiliary principle technique
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General regularized nonconvex mixed equilibrium problems: iterative methods by auxiliary principle technique Javad Balooee1 · Yeol Je Cho2 Received: 31 May 2019 / Accepted: 24 October 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract This paper introduces a new class of equilibrium problems named general regularized nonconvex mixed equilibrium problem. By using the auxiliary principle technique, some predictor–corrector methods for solving such class of general regularized nonconvex mixed equilibrium problems are suggested and analyzed. The study of convergence analysis of the proposed iterative algorithms requires either jointly pseudomonotonicity or partially mixed relaxed and strong monotonicity property of the operator and bifunction involved in the considered problem. As a consequence of our main results, we provide the correct versions of the algorithms and results presented in the literature. Keywords Generalized regularized nonconvex mixed equilibrium problems · Auxiliary principle technique · Prox-regularity · Predictor–corrector methods · Nonconvex sets · Convergence analysis Mathematics Subject Classification 47H05 · 47J20 · 47J25 · 49J40 · 65K10 · 65K15
1 Introduction During the last decade, theory of existence of solutions and theory of solution methods for variational inequalities and equilibrium problems have been investigated in the setting of nonconvex sets, namely, uniformly prox-regular sets, see, for example, [3,8,10,12,15,23] and the references therein. It is well-known that the equilibrium problem is a unified model of several problems, namely, variational inequality problems, complementarity problems, fixed point problems, saddle point problems, optimization problems, etc; See, for example, [1–3,6,11,13,14,21,27] and the references therein.
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Javad Balooee [email protected] Yeol Je Cho [email protected]
1
Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran
2
Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea
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J. Balooee, Y. J. Cho
Recently, Noor [24] introduced and studied mixed quasi regularized variational inequalities in the setting of uniformly prox-regular sets. He used the auxiliary principle technique to suggest and analyze some iterative algorithms for solving such class of mixed quasi regularized variational inequalities. He studied the convergence analysis of the proposed algorithms under pseudomonotonicity and partially relaxed strong monotonicity of the operator involved in the formulation of the problem. The purpose of this paper is in twofold. The first objective is to introduce a general regularized nonconvex mixed equilibrium problem (in short, GRNMEP) which include mixed quasi regularized variational inequalities [24], equilibrium problems, general variational inequalities [26], etc, as special cases. By using auxiliary principle technique, we propose some predictor–corrector methods for solving the GRNMEP. The convergence of the sequences generated by the proposed method
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