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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]

Nonlinear complementarity problems (NCP) have been extensively studied in optimization due to its widespread applications. In this paper, we utilize the neural dynamic approach to solve the NCP. By integrating the famous FB function an NR function, we construct a new type of complementarity functions with one parameter p , which is aesthetically pleasing and easy to apply. Combined with the Lagrange multiplier method, a new type of merit function is also developed. Based on the complementarity function and merit function, we transform the NCP into an unconstrained minimization problem. Then, by KKT condition and gradient descent method, we propose a Lagrange neural network method. Under mild conditions, every equilibrium point of the proposed neural network model is a solution of the NCP. More importantly, by throwing a delay factor, we also develop a novel delayed neural network model. Both of these networks are shown to be global convergent, Lyapunov stable and exponential stable. Finally, we give some numerical experiments of the two neural network approaches and also give some applications to the compressed sensing signal reconstruction. Simulation results indicate that the parameter p in the complementarity function plays an important role on the convergence rate of the two neural networks. The delayed neural network outperforms the non-delayed neural network in some specific situations. It also demonstrates that the two networks can efficiently reconstruct the original signals. • A new type of complementary functions is constructed. • A new neural network method is proposed. • We propose a delayed neural network and give its complete convergence analysis. • The two neural networks are applied to solve the CS efficiently. [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]