Mean-field forward-backward stochastic variational inequalities with oblique subgradients and nonlocal parabolic variational inequalities

Mobility-as-a-service (MaaS) system as a multi-leader-multi-follower game: A single-level variational inequality (VI) formulation

Evolution variational inequalities with general costs

A generalized system of triple fractional stochastic differential variational inequalities with Lévy jumps

Adaptive step-size extragradient algorithm with variance reduction for solving mixed variational inequality problems

Inertial subgradient–extragradient schemes for variational inequalities in 2-uniformly convex spaces

We investigate the asymptotic behavior of incompressible thermally coupled Bingham flows in a three-dimensional thin domain Ω ε , subject to Tresca slip conditions on part of the boundary and Dirichlet conditions elsewhere. Using a variational inequality framework and a rescaling of the thin domain onto a fixed reference domain, we establish the existence and uniqueness of weak solutions together with uniform ε-independent estimates. By combining compactness arguments and asymptotic analysis as ε → 0 , we derive the effective limiting model and identify the asymptotic form of the Tresca slip condition. In particular, we prove that the pressure becomes independent of the transverse variable and obtain a Reynolds-type equation governing the limit flow.

In some situations, drivers prefer to select the most reliable paths (MRP). In the minimum-delay routing problem in communication networks, signals need to be transmitted on the MRPs. Since a minimum cost path may not be an MRP, the traditional minimum cost path based user equilibrium does not apply to such scenarios. Therefore, the study of the connectivity reliability based user equilibrium (RUE) model which is based on the MRPs is needed. Unfortunately, none of the previous studies ever covered the RUE based on the MRPs. In this paper, a RUE variational inequality (VI) model based on the MRPs is presented. The path reliability is the multiplication of the reliability of the links on the path. A heuristic algorithm is presented to solve the VI model. A multiplication-based algorithm is used to find the MRP in the solution process. The model and algorithm are applied to two networks and their validity is shown. The method presented in this paper is useful for the emergency evacuation on transportation networks or the minimum-delay routing problem in communication networks.

We devise a relaxed-inertial subgradient extragradient method for solving pseudomonotone equilibrium problems on Hadamard manifolds. The method uses a self-adaptive step size that does not depend on the Lipschitz constant. Under suitable conditions, we demonstrate that every sequence produced by the proposed algorithms converges to a solution of the equilibrium problem. We also prove linear convergence of the proposed method when the bifunction satisfies strong pseudomonotonicity and Lipschitz continuity. We then provide an application of our main result to variational inequality and optimization problems using the proposed algorithms. We conclude our paper with a numerical example to verify our convergence result.

This paper presents an alternated inertial subgradient extragradient projection method for solving variational inequalities in real Hilbert spaces. The proposed algorithm employing an adaptive halfspace correction parameter at each iteration, has the advantage of improving stability and enhancing convergence. Under suitable conditions, weak convergence and strong convergence are established. Numerical experiments validate the effectiveness of the proposed method compared against existing related algorithms.

Solving variational inequalities using an algorithm with extrapolations from the past and relaxation

A positivity preserving and highly scalable framework for the variational inequality solution of the transport equation

The aim of this study is to analyze a mathematical model describing the contact between a deformable body and an obstacle. The material behavior is governed by a linear viscoelastic Kelvin-Voigt constitutive law involving time-fractional derivatives. The contact model combines normal compliance with unilateral stress constraints under Coulomb's friction law. We first derive the variational formulation and prove existence of weak solutions. Furthermore, we analyze a penalized version of the problem without constraints, demonstrating that its solutions converge to the original variational inequality solution as the penalty parameter tends to zero.

Complex-valued bidirectional associative memory (BAM) neural networks with fractional-order dynamics and delays can exhibit transient instabilities that degrade synchronisation and short-horizon predictability. This paper develops a unified analytical and data-driven framework to assess stability, synchronisation, and predictability in such networks. First, using Caputo fractional calculus and Lyapunov-Mittag-Leffler techniques, we derive sufficient conditions for global Mittag–Leffler synchronisation of a drive-response BAM pair under a linear error-feedback controller and obtain an explicit time-to-tolerance bound. Second, to quantify local transient instability from finite trajectory data, we propose a micro-ensemble finite-time Lyapunov exponent (FTLE) estimator based on the geometric-mean growth of small perturbations over short windows, avoiding variational equations. We further introduce k-nearest-neighbour prediction-error Lyapunov proxies (full-state and modulus-based) to connect local instability to forecasting performance. Numerical experiments on fractional-order complex-valued BAM benchmarks confirm effective synchronisation under the proposed control and demonstrate a clear correspondence between FTLE levels and prediction errors. The resulting framework provides practical, reproducible diagnostics for complex-valued neural systems in data-limited settings.

In this paper, we establish the connections between the solutions of some classes of vector-type variational control inequalities, denoted by (V CI) and (WV CI), and (local, weak) quasi-efficient solutions of the associated multiobjective optimization problem, denoted by (P). In this regard, we use local (strictly) approximately star-shaped and/or local approximately pseudo-convex integral functionals. In addition, some illustrative examples are also presented to verify the statements and established theoretical results.

This study presents a unified analytical–numerical framework for the El Borhamy–Rashad–Sobhy equation with Duffing-type nonlinearity, augmented by an external harmonic forcing term. The used application in the study is motivated by the electromechanical dynamics of salient-pole synchronous machines. Starting from an energy-based formulation, the harmonic balance method is used to obtain closed-form frequency-response relations. The local stability of the periodic orbit is further quantified via a Floquet-based test derived from the linear variational equation, yielding stability-classified frequency-response curves. While the method of multiple scales captures primary, superharmonic, and subharmonic resonance conditions and the associated stability boundaries are analyzed. Beyond the weakly nonlinear regime, numerical bifurcation analysis is performed to trace a period-doubling route to chaos, supported by Poincaré sections, the largest Lyapunov exponent, and time-series diagnostics. The system is further coupled with linear and nonlinear piezoelectric energy-harvesting branches, to demonstrate that, the large-amplitude responses enhance harvested power and the cubic stiffness can be tuned for bandwidth optimization. Overall, the results bridge perturbation theory with nonlinear time-domain diagnostics, providing design-relevant insights for broadband energy harvesters and electromechanical systems operating under parametric excitation and external forcing.

In this paper, we introduce an Anderson-accelerated method with a self-adaptive step size to solve variational inequality problem with a pseudo-monotone operator. The sequence generated by the proposed algorithm converges weakly to a solution of the variational inequality problem under some suitable conditions. Finally, some numerical experiments are performed to show the effectiveness of the method.

This paper introduces a support condition approach for mixed insurance risk control using linear programming inspired by occupation measures. The model includes reinsurance – continuous control, capital injection, and dividends – singular controls. Building upon [Goreac et al. (2022). Linearization techniques and the dual algorithm for a class of mixed singular/continuous control problems in reinsurance. Part I: Theoretical aspects. Applied Mathematics and Computation, 431, Article 127321], we propose: (i) a dual formulation akin to Fenchel-Legendre transforms that characterizes optimal measure support and enables efficient policy extraction once the value function is known; (ii) validation against benchmarks from [Goreac et al. (2024). Linearization techniques and the dual algorithm for a class of mixed singular/continuous control problems in reinsurance. Part II: Numerical aspects. Applied Mathematics and Computation, 473, Article 128655], including exponential claims without reinsurance and general distributions with reinsurance but no injections. Numerical results recover threshold-based dividend and capital injection policies for various claim distributions, demonstrating the method's efficiency in solving Hamilton–Jacobi–Bellman variational inequalities via measure relaxation.

Abstract The one-dimensional dynamic contact problem describing a rigid obstacle collided by an elastic bar is considered in gravitational field. The collision problem is non-smooth with respect to velocity and axial strain, and formulated as a variational inequality. For the corresponding wave equation subject to complementary conditions which are imposed at the contact boundary, its nonlinear solution is expressed with the help of Fourier Series representing longitudinal vibrations. Moreover, before the bar rebound, an analytical solution comprising piece-wise quadratic function is constructed on a partition of the rectangular space-time domain along characteristics. The analytical benchmark is implemented for low initial speeds in numerical experiments solving the variational inequality over uniform space-time triangulation with the Space-Time Primal-Dual Active Set method of semi-smooth Newton type.

Approximate solutions to vector variational inequalities based on improvement sets: Existence, characterization, and optimality conditions