An Adaptive Geodesic-Relaxed Extragradient Method for Equilibrium and Fixed-Point Problems in CAT(0) Spaces

Abstract

We propose an adaptive geodesic-relaxed extragradient method for finding a common solution of a pseudomonotone equilibrium problem and a fixed-point problem in a complete CAT(0) space. The method combines a two-point geodesic memory step, two regularized equilibrium subproblems, an adaptive stepsize rule, and a fixed-point correction. The terminology ``geodesic-relaxed'' is used deliberately: the memory point is a geodesic interpolation of two successive iterates and is not an extrapolation beyond the current iterate. The CAT(0) geometry yields the strong convexity of squared distance needed to derive the three-point inequalities underlying the extragradient descent estimate. The adaptive stepsizes are shown to remain bounded away from zero without requiring the Lipschitz-type constants in the implementation. Boundedness follows for quasi-nonexpansive fixed-point mappings. When the fixed-point mapping is a contraction, the whole sequence converges strongly to the unique common solution, and the equilibrium and fixed-point residuals vanish. For the broader nonexpansive case, a cluster-point result is stated under an explicit asymptotic-regularity and \(\Delta\)-demiclosedness hypothesis. A reproducible one-dimensional experiment with a nontrivial contractive fixed-point map illustrates the theoretical behavior without claiming universal acceleration.