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In this paper, a modulus-based Shamanskii-Like Levenberg-Marquardt method is proposed for solving nonlinear complementarity problems (NCPs). First, the NCP is reformulated in the form of an equivalent non-smooth system of equations. Then, a non-smooth Shamanskii-Like Levenberg-Marquardt method using a non-monotone r-order Armijo line search is developed by generalizing a smooth Levenberg-Marquardt method to solve the resulting system. Global convergence of the proposed method is achieved under some suitable assumptions. Numerical experiments verify the feasibility and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This paper presents a Mehrotra-type predictor–corrector algorithm for monotone nonlinear complementarity problems over symmetric cones. The proposed algorithm is a second-order predictor–corrector interior point algorithm. Based on a one-norm wide neighbourhood, the iteration complexity of the algorithm is estimated and some numerical results are provided. The numerical results show that the algorithm is efficient and reliable. [ABSTRACT FROM AUTHOR]

We enrich the theory of variational inequalities in the case of an aggregative structure by implementing recent results obtained by using the Selten–Szidarovszky technique. We derive existence, semi-uniqueness and uniqueness results for solutions and provide a computational method. As an application we derive very powerful practical equilibrium results for Nash equilibria of sum-aggregative games and illustrate with Cournot oligopolies. [ABSTRACT FROM AUTHOR]

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 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. [ABSTRACT FROM AUTHOR]

In this paper, we establish a nonlinear complementarity model and algorithm for supply chain equilibrium management problem consisting of manufacturers, retailers and consumer markets. This work focus on the price of the goods of retailer sell to consumer market in which is a function of the amount of products that are transacted between the retailer and the consumer. Based on this, we investigate the optimizing behavior of the various decision-makers, derive the equilibrium conditions of the manufacturers, the retailers and the consumer markets respectively, and establish a nonlinear complementarity model of this problem. To obtain optimal decision for the problem, we propose a new type of algorithm based on established model, and its global convergence is presented without the assumption of global Lipschitz continuous in detail. The efficiency of given algorithm is also illustrated through some numerical examples. [ABSTRACT FROM AUTHOR]

• A general algorithm for multisurface plasticity is presented. • A general frame to define sequential linear complementarity problem for plasticity. • The set of active constraints is naturally identified by Lemke's algorithm. • Stress is updated using the method of sequential linear complementarity problem. • The robustness of the algorithm is demonstrated via the 3D isoerror map. In this paper, a general algorithm is presented for the integration of multisurface plasticity models. The algorithm combines the return mapping with the technique of the sequential linear complementarity problem (SLCP) to update stress. The set of active constraints is naturally identified by the classical Lemke's algorithm. With the assumptions of isotropic linear elasticity, perfect plasticity, and the associated flow rule, all details are provided in matrix notations to facilitate computer implementation. The extension to hardening/softening multisurface plasticity models is also presented. The application of the algorithm is demonstrated via simulation of three types of geotechnical problems in 2D and 3D. Both linear and nonlinear multisurface plasticity models, e.g. Mohr-Coulomb and generalized Hoek-Brown yield criteria, are examined within the framework of the proposed algorithm. The numerical results are in good agreement with the analytic solutions. Moreover, the accuracy of the proposed stress integration procedure is investigated through 3D isoerror map. The convergence using the consistent tangent matrix at the global level is examined by a one-increment example consisting of one element. [ABSTRACT FROM AUTHOR]

In this paper we develop a numerical method for pricing American options under regime-switching model, whose solutions are option buyer indifference prices. The problem is formulated as a nonlinear complementarity problem. We apply interior penalty method to approximate the differential complementarity problem, which results in a system of one degenerate parabolic equation and one ordinary differential equation, weakly coupled by nonlinear exponential term. We formulate the problems in variational form in appropriate spaces and prove comparison principle. Then, as a consequence, we derive estimates about the convergence rate of the interior penalty method. We discretize the penalized problem by fully implicit finite difference scheme and prove that the early exercise constraint is strictly satisfied. We establish comparison principle, uniqueness and boundedness of the numerical solution. To solve the nonlinear system of algebraic equations we use Newton method and partial Newton method. We present numerical results that confirm the theoretical statements. To improve the computational efficiency, we unfold the two-grid idea, combining both iteration processes – Newton method and partial Newton method. We illustrate the efficiency of this approach. [ABSTRACT FROM AUTHOR]

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. • Novel algorithmic formulations for rate-independent crystal plasticity theory based on incremental energy minimization. • Algorithms can also be applied to conventional, non-variational plasticity models. • Algorithms can be applied to local as well as to gradient-enhanced plasticity models. • Algorithm guarantees a regular tangent matrix and identification of active slip systems. • Reinterpretation of active-set algorithms as nonlinear complementarity problem (NCP). [ABSTRACT FROM AUTHOR]

For constrained equations with nonisolated solutions and a certain family of Newton-type methods, it was previously shown that if the equation mapping is 2-regular at a given solution with respect to a direction which is interior feasible and which is in the null space of the Jacobian, then there is an associated large (not asymptotically thin) domain of starting points from which the iterates are well defined and converge to the specific solution in question. Under these assumptions, the constrained local Lipschitzian error bound does not hold, unlike the common settings of convergence and rate of convergence analyses. In this work, we complement those previous results by considering the case when the equation mapping is 2-regular with respect to a direction in the null space of the Jacobian which is in the tangent cone to the set, but need not be interior feasible. Under some further conditions, we still show linear convergence of order 1/2 from a large domain around the solution (despite degeneracy, and despite that there may exist other solutions nearby). Our results apply to constrained variants of the Gauss–Newton and Levenberg–Marquardt methods, and to the LP-Newton method. An illustration for a smooth constrained reformulation of the nonlinear complementarity problem is also provided. [ABSTRACT FROM AUTHOR]

Having in mind singular solutions of smooth reformulations of complementarity problems, arising unavoidably when the solution in question violates strict complementarity, we study the behavior of Newton-type methods near singular solutions of nonlinear equations, assuming that the operator of the equation possesses a strongly semismooth derivative, but is not necessarily twice differentiable. These smoothness restrictions give rise to peculiarities of the analysis and results on local linear convergence and asymptotic acceptance of the full step, the issues addressed in this work. Moreover, we consider not only the basic Newton method, but also some stabilized versions of it intended for tackling singular (including nonisolated) solutions. Applications to nonlinear complementarity problems are also dealt with. [ABSTRACT FROM AUTHOR]

Prediction of wave-induced instantaneous (oscillatory or momentary) liquefaction is particularly important for the design of offshore foundations. Most previous studies applied the linear Darcy model to characterize the porous flow in a seabed. This treatment was found to cause fallacious tensile stresses in a non-cohesive seabed. In this study, to overcome such shortcomings of previous models, a non-Darcy flow model is proposed based on a Karush–Kuhn–Tucker (KKT) condition. In the KKT condition, the primal constraint arises from the fact that the tensile behavior does not exist in a non-cohesive seabed, while the dual condition arises from the physical evidences that the pore-fluid velocity increases during liquefaction. The non-linearity of the present model is handled by the Newton–Raphson method within the standard finite element framework, without coding constrained variational principle. This highlights the convenience for numerical implementation. The difficulties in treating the nonlinearity by previous dynamic permeability model are also eliminated by the non-Darcy flow model. The merits of the proposed model are validated by examining four numerical treatments and two liquefaction criteria. The liquefaction depth by the present model is found to be roughly 0.73 times of the value by the linear Darcy model. • A non-Darcy flow model for wave-induced instantaneous liquefaction is proposed. • This model is based on a KKT condition imposed by the penalty method. • The non-physical tensile behavior caused by constant permeability is removed. • Numerical accuracy, algorithm stability and implementing convenience are achieved. [ABSTRACT FROM AUTHOR]

The estimation of the wave-induced instantaneous liquefaction is particularly important for the design of foundations of offshore structures. Regarding the occurrence of liquefaction in a non-cohesive seabed, most existing studies using constant permeability were found to cause fallacious tensile stresses in the liquefied zone and further pollute the overall pore pressure distribution. A dynamic permeability model was previously presented to mitigate the shortcoming but posed difficulties in the nonlinear convergence. To overcome the shortcoming of the previous studies, this study proposes the concept of modelling the liquefaction-involved wave-seabed interactions as a nonlinear complementarity problem, wherein a Karush–Kuhn–Tucker condition is constructed, based on revisiting the liquefaction criterion most widely applied in ocean engineering. The Lagrange multiplier method and the primal–dual active set strategy are employed to numerically deal with the nonlinear complementarity problem. The performance of the chosen multiplier space is investigated by theoretical analyzing and numerical modelling. Compared with the previous dynamic permeability model, the present model is totally free of extra parameters and precisely fulfills the no-tension requirement. Moreover, the difficulties of dynamic permeability in the nonlinear convergence are overcome and no divergence is observed in the numerical tests. [ABSTRACT FROM AUTHOR]