Proximal and Contraction method with Relaxed Inertial and Correction Terms for Solving Mixed Variational Inequality Problems
Chidi Elijah Nwakpa, Austine Efut Ofem, Kalu Okam Okorie, Chinedu Izuchukwu, Chibueze Christian Okeke
TL;DR
This work addresses solving convex mixed variational inequality problems (MVI) in real Hilbert spaces by proposing a proximal and contraction method augmented with a relaxed inertial term and two correction terms, along with a self-adaptive stepsize. The algorithm generates sequences using $w_n$, $y_n=prox_{\lambda_n g}(w_n-\lambda_n\mathcal{T}w_n)$, $d(w_n,y_n)$, $z_n$, and $x_{n+1}=(1-\theta)w_n+\theta z_n$, ensuring weak convergence to the solution set $\Delta(\mathcal{T}; g)$ under generalized monotonicity and standard assumptions; the method collapses to previous approaches when the correction and relaxation parameters vanish. Numerical experiments in MATLAB compare the proposed method to several existing schemes, showing improved convergence speed and lower CPU time due to the inertial and two-correction framework across multiple problem dimensions. The results highlight the practical benefits of the relaxation, inertial extrapolation, and correction terms in accelerating proximal-contraction schemes for MVIs. This approach offers a scalable, projection-efficient alternative for solving MVIs with potential applications in various scientific and engineering domains.
Abstract
We propose in this paper a proximal and contraction method for solving a convex mixed variational inequality problem in a real Hilbert space. To accelerate the convergence of our proposed method, we incorporate an inertial extrapolation term, two correction terms, and a relaxation technique. We therefore obtain a weak convergence result under some mild assumptions. Finally, we present numerical examples to practically demonstrate the effectiveness of the relaxation technique, the inertial extrapolation term, and the correction terms in our proposed method.