Convergence Rate of A Unified Prediction-Correction Algorithm for Linearly Constrained Equilibrium Problems

Equilibrium problem is an important mathematical model, which provides a unified framework of variational inequalities, complementarity problem, optimization problem, minimax problem and fixed point problem as special cases. Auxiliary principle is crucial tool for designing algorithms of equilibrium problems with abstract feasible set, which generally involve the construction of auxiliary equilibrium problems via a positive-definite matrix. In this paper, a unified prediction-correction algorithm based on auxiliary principle is proposed for solving monotone equilibrium problems (MEP) with set constraint and linear constraints. The saddle point problem associated with the Lagrangian function of MEP is equivalently characterized by a mixed equilibrium problem. Moreover, an auxiliary equilibrium problem is introduced by the mixed equilibrium problem and a positive definite matrix. Some characterizations, such as nonemptiness, closedness, convexity and firmly non-expansiveness, for solutions set of the auxiliary equilibrium problem are established. Then a unified prediction-correction algorithm is suggested based on the auxiliary equilibrium problem. The convergence results of the proposed algorithms are established under some mild assumptions. We also obtain the sublinear convergence rate of the proposed algorithms in both ergodic and nonergodic senses. Some special positive-definite matrix $${\textbf{D}}$$ D in auxiliary equilibrium problem are also suggested, under which the proposed algorithms recover the existed methods. Finally, some numerical examples are reported to show the feasibility and validity of the proposed algorithms.