An Inertial Accelerated Predictor–Corrector Algorithm for Exponential Variational Inequalities with Sharp Convergence Guarantees and Applications

Abstract

We study exponentially convex nonlinearities and exponentially monotone operators in Hilbert spaces for exponential variational inequalities (EVIs). We show that the exponential nonlinearity is the gradient of a convex functional, so it is monotone. We use this fact to prove existence, uniqueness, and contractivity of the associated fixed-point mapping for every strong monotonicity constant α>0 of the underlying operator. This task is performed using an explicit, computable step-size range and linear convergence rate. We present a two-step inertial predictor–corrector technique and prove its strong convergence. Applying the framework to the Bratu obstacle problem (which models thermal ignition in combustion systems) yields the explicit, sharp requirement (b−a)2eM<1 for contractivity of the associated fixed-point map and, hence, for existence, uniqueness, and convergence in this application. When this requirement is satisfied, numerical experiments demonstrate a noticeable acceleration of up to 5% from the inertial approach in the contractive regime’s mid-range. Neither the non-contractive regime nor the smallest tested domain length showed any obvious improvement.