Nonlinear Complementarity Problem Research Articles

Finite Elements with Switch Detection for direct optimal control of nonsmooth systems

This paper introduces Finite Elements with Switch Detection (FESD), a numerical discretization method for nonsmooth differential equations. We consider the Filippov convexification of these systems and a transformation into dynamic complementarity systems introduced by Stewart (Numer Math 58(1):299–328, 1990). FESD is based on solving nonlinear complementarity problems and can automatically detect nonsmooth events in time. If standard time-stepping Runge–Kutta (RK) methods are naively applied to a nonsmooth ODE, the accuracy is at best of order one. In FESD, we let the integrator step size be a degree of freedom. Additional complementarity conditions, which we call cross complementarities, enable exact switch detection, hence FESD can recover the high order accuracy that the RK methods enjoy for smooth ODE. Additional conditions called step equilibration allow the step size to change only when switches occur and thus avoid spurious degrees of freedom. Convergence results for the FESD method are derived, local uniqueness of the solution and convergence of numerical sensitivities are proven. The efficacy of FESD is demonstrated in several simulation and optimal control examples. In an optimal control problem benchmark with FESD, we achieve up to five orders of magnitude more accurate solutions than a standard time-stepping approach for the same computational time.

Solving forward and inverse problems of contact mechanics using physics-informed neural networks

This paper explores the ability of physics-informed neural networks (PINNs) to solve forward and inverse problems of contact mechanics for small deformation elasticity. We deploy PINNs in a mixed-variable formulation enhanced by output transformation to enforce Dirichlet and Neumann boundary conditions as hard constraints. Inequality constraints of contact problems, namely Karush–Kuhn–Tucker (KKT) type conditions, are enforced as soft constraints by incorporating them into the loss function during network training. To formulate the loss function contribution of KKT constraints, existing approaches applied to elastoplasticity problems are investigated and we explore a nonlinear complementarity problem (NCP) function, namely Fischer–Burmeister, which possesses advantageous characteristics in terms of optimization. Based on the Hertzian contact problem, we show that PINNs can serve as pure partial differential equation (PDE) solver, as data-enhanced forward model, as inverse solver for parameter identification, and as fast-to-evaluate surrogate model. Furthermore, we demonstrate the importance of choosing proper hyperparameters, e.g. loss weights, and a combination of Adam and L-BFGS-B optimizers aiming for better results in terms of accuracy and training time.

Numerical investigation of the wheel-rail interaction in a stacked multibody system with multipoint frictional contact dynamics under strong earthquakes

Stacked multibody systems, exemplified by overhead cranes in nuclear power plants (OCNPP), play a crucial role in engineering applications. Under strong earthquakes, non-smooth behaviors, including detachment and stick-slip between wheels and rails, may induce uncertain motions, potentially triggering catastrophic accidents. In this study, a co-simulation method using ANSYS and MATLAB is proposed for modeling the stacked components. The wheel-rail relationship is modeled with multiple frictional unilateral constraints, which are delineated by a set-valued force law. The resultant contact-impact problem is formulated as a nonlinear complementarity problem, and then reconstructed and rescaled using a model smoothing method combined with a normalization technique to facilitate convergence. Lastly, numerical simulations are conducted, taking into account various wheel locking conditions, to explore the nonlinear behavior and stress distribution. The results illustrate the non-smooth characteristics of frictional contact within the stacked components. The presented work is expected to guide the structural optimization of stacked multibody systems.

General convergence results of the modulus-based methods for vertical nonlinear complementarity problems

General convergence results of the modulus-based methods for vertical nonlinear complementarity problems

Modulus-based synchronous multisplitting method for horizontal nonlinear complementarity problem

Modulus-based synchronous multisplitting method for horizontal nonlinear complementarity problem

A novel predefined-time neurodynamic approach for mixed variational inequality problems and applications

In this paper, we propose a novel neurodynamic approach with predefined-time stability that offers a solution to address mixed variational inequality problems. Our approach introduces an adjustable time parameter, thereby enhancing flexibility and applicability compared to conventional fixed-time stability methods. By satisfying certain conditions, the proposed approach is capable of converging to a unique solution within a predefined-time, which sets it apart from fixed-time stability and finite-time stability approaches. Furthermore, our approach can be extended to address a wide range of mathematical optimization problems, including variational inequalities, nonlinear complementarity problems, sparse signal recovery problems, and nash equilibria seeking problems in noncooperative games. We provide numerical simulations to validate the theoretical derivation and showcase the effectiveness and feasibility of our proposed method.

The simplified modulus-based matrix splitting iteration method for the nonlinear complementarity problem

<abstract><p>In this paper, the simplified modulus-based matrix splitting iteration method was extended to solve the nonlinear complementarity problem, and the convergence conditions were presented from the spectral radius and the matrix norm. Then, for the special cases of this method, we provided some concrete convergence conditions as well as the quasi-optimal parameter matrix. Moreover, some numerical examples were illustrated to show the validity of the convergence results.</p></abstract>

A two-step iteration method for solving vertical nonlinear complementarity problems

A two-step iteration method for solving vertical nonlinear complementarity problems

Power Dissipation and Wear Modeling in Wheel–Rail Contact

This paper is concerned with the modeling of power dissipation due to friction and its relation with wear estimation in wheel–rail contact. Wear is a complex multi-scale and multi-physical phenomenon appearing in rolling contact. Wear is generated by high contact stress and the work of friction forces. This phenomenon leads to the appearing of the worn material in the form of wear debris between contacting surfaces. In contact models, wear is usually described in terms of the wear depth function. This function modifies the gap between the contacting bodies as well as the shape of the surfaces of the wheel and rail in contact. In this paper, besides the wear depth function, the dissipated energy, rather than the contact stress, is taken into account to evaluate the wear impact on rail or wheel surfaces. The dissipated energy allows us to more precisely evaluate the wear debris amount as well as the depth of wear and its distribution along the contact interface. A two-dimensional rolling contact problem with frictional heat flow is considered. The elasto-plastic deformation of the rail is considered. This contact problem is governed by a coupled system of mechanical and thermal equations in terms of generalized stresses, displacement and temperature. The finite element method is used to discretize this problem. A discretized system of equations with nonpenetration and friction conditions is transformed and formulated as a nonlinear complementarity problem. The generalized Newton method is applied to numerically solve this mechanical subproblem. The Cholesky method is used to find the solution of the heat-conductive problem. The dissipated power is evaluated based on the resultant force and slip at a reference point. Numerical results including the distribution of slip velocity, power factor and wear rate are provided and discussed.

Rate-independent gradient-enhanced crystal plasticity theory — Robust algorithmic formulations based on incremental energy minimization

Numerically robust algorithmic formulations suitable for rate-independent crystal plasticity are presented. They cover classic local models as well as gradient-enhanced theories in which the gradients of the plastic slips are incorporated by means of the micromorphic approach. The elaborated algorithmic formulations rely on the underlying variational structure of (associative) crystal plasticity. To be more precise and in line with so-called variational constitutive updates or incremental energy minimization principles, an incrementally defined energy derived from the underlying time-continuous constitutive model represents the starting point of the novel numerically robust algorithmic formulations. This incrementally defined potential allows to compute all variables jointly as minimizers of this energy. While such discrete variational constitutive updates are not new in general, they are considered here in order to employ powerful techniques from non-linear constrained optimization theory in order to compute robustly the aforementioned minimizers. The analyzed prototype models are based on (1) nonlinear complementarity problem (NCP) functions as well as on (2) the augmented Lagrangian formulation. Numerical experiments show the numerical robustness of the resulting algorithmic formulations. Furthermore, it is shown that the novel algorithmic ideas can also be integrated into classic, non-variational, return-mapping schemes.

An Incremental Gradient Method for Optimization Problems With Variational Inequality Constraints

We consider minimizing a sum of agent-specific nondifferentiable merely convex functions over the solution set of a variational inequality (VI) problem in that each agent is associated with a local monotone mapping. This problem finds an application in computation of the best equilibrium in nonlinear complementarity problems arising in transportation networks. We develop an iteratively regularized incremental gradient method where at each iteration, agents communicate over a directed cycle graph to update their solution iterates using their local information about the objective and the mapping. The proposed method is single-timescale in the sense that it does not involve any excessive hard-to-project computation per iteration. We derive non-asymptotic agent-wise convergence rates for the suboptimality of the global objective function and infeasibility of the VI constraints measured by a suitably defined dual gap function. The proposed method appears to be the first fully iterative scheme equipped with iteration complexity that can address distributed optimization problems with VI constraints over cycle graphs.

A modified multivariate spectral gradient projection method for nonlinear complementarity problems

A modified multivariate spectral gradient projection method for nonlinear complementarity problems

A Two-Step Newton Algorithm for the Weighted Complementarity Problem with Local Biquadratic Convergence

We discuss the weighted complementarity problem, extending the nonlinear complementarity problem on Rn. In contrast to the NCP, many equilibrium problems in science, engineering, and economics can be transformed into WCPs for more efficient methods. Smoothing Newton algorithms, known for their at least locally superlinear convergence properties, have been widely applied to solve WCPs. We suggest a two-step Newton approach with a local biquadratic order convergence rate to solve the WCP. The new method needs to calculate two Newton equations at each iteration. We also insert a new term, which is of crucial importance for the local biquadratic convergence properties when solving the Newton equation. We demonstrate that the solution to the WCP is the accumulation point of the iterative sequence produced by the approach. We further demonstrate that the algorithm possesses local biquadratic convergence properties. Numerical results indicate the method to be practical and efficient.

Dynamic equilibrium analyses in a ride-sourcing market under travel time uncertainty

This paper proposes a dynamic equilibrium framework for the morning commute problem in the ride-sourcing market under travel time uncertainty. It is formulated as a simultaneous departure time and mode choice problem (SDTMCP) with random travel delay. Commuters are considered to choose among four travel modes: (1) driving alone, (2) taking non-shared service, (3) taking ride-sharing service, and (4) taking the metro. The equilibrium state is reached when no commuter can reduce their expected travel cost by unilaterally changing their mode or departure time. We derive the dynamic equilibrium queuing patterns analytically under certain assumptions, and develop an approach using nonlinear complementarity problem (NCP) to solve general problems where analytical results cannot be readily derived. We further explore the pricing strategy of the ride-sourcing platform, in which travel time uncertainty and the prices of ride-sourcing services jointly determine the profit performance. A mathematical program with equilibrium constraints (MPEC) is formulated to search for the optimal service prices to achieve the maximum profit. The framework is then extended to investigate the pricing strategy in a two-sided ride-sourcing market and examine the metro crowdedness effect in the numerical studies.

A Semidefinite Relaxation Method for Linear and Nonlinear Complementarity Problems with Polynomials

A Semidefinite Relaxation Method for Linear and Nonlinear Complementarity Problems with Polynomials

A modulus-based matrix splitting method for the vertical nonlinear complementarity problem

A modulus-based matrix splitting method for the vertical nonlinear complementarity problem

A Mixed Equilibrium Model and Optimal Path Platooning Method for CAV Platoons in Heterogeneous Traffic Flow

As the emergence of the connected and autonomous vehicles (CAVs), the platooning technology is believed to play a key role in the future intelligent transportation system. However, current studies mainly focus on the beneficial sides of CAV platoons, and less attention is given to their negative effects. This study develops a mixed equilibrium model for CAV platoons and human-driven vehicles (HDVs), which consider both the positive and negative sides of CAV platooning. On the positive side, CAV platoons are assumed to follow user equilibrium (UE) route choice for their information advantages, while HDVs to follow stochastic user equilibrium (SUE). CAV platoons are also presumed to improve the road capacity. On the negative side, the speed of CAV platoons is slower than that of HDVs for safety stakes, which will impede the latter to overtake. The HDVs is split up into overtaking and nonovertaking flows with different speeds. Furthermore, the model is built up as a mixed UE-SUE equilibrium problem and reformulated as a nonlinear complementarity problem. In addition, an optimal path platooning method is proposed to reduce the negative effects, by integrating travel costs of both CAV platoons and HDVs into its objective function. Numerical results show that the introduction of CAV platoons may increase the travel cost at the initial stage, and the proposed method can effectively reduce the platooning disturbance, thus helps promoting the wider applications of CAV platoons.

A smoothing Newton method based on the modulus equation for a class of weakly nonlinear complementarity problems

A smoothing Newton method based on the modulus equation for a class of weakly nonlinear complementarity problems

A Finsler Geometrical Programming Approach to the Nonlinear Complementarity Problem of Traffic Equilibrium

We consider a geometrical approach to the optimisation problems motivated by transportation system management. Here, we provide a comprehensive account of geometric programming based on the elementary Finsler geometry in {mathbb {R}}^n. Then, we present a Finslerian dynamical model for the nonlinear complementarity problem of traffic equilibrium that can be applied to a variety of equilibrium problems.

Inexact fixed-point iteration method for nonlinear complementarity problems

Based on the modulus decomposition, the structured nonlinear complementarity problem is reformulated as an implicit fixed-point system of nonlinear equations. Distinguishing from some existing modulus-based matrix splitting methods, we present a flexible modulus-based inexact fixed-point iteration method for the resulting system, in which the subproblem can be solved approximately by a linear system-solver. The global convergence of the proposed method is established by assuming that the system matrix is positive definite. Some numerical comparisons are reported to illustrate the applicability of the proposed method, especially for large-scale problems.