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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]

This paper establishes a theoretical framework of infeasible Mehrotra-type predictor-corrector algorithm for Cartesian nonlinear complementarity problems over symmetric cones which can be regarded as an extension of Liu et al. [Polynomial convergence of Mehrotra-type predictor-corrector algorithm for the Cartesian -LCP over symmetric cones, Optimization 64(4) (2013), pp. 1-23] for Cartesian linear complementarity problems over symmetric cones to Cartesian nonlinear complementarity problems over symmetric cones. The iteration complexities of the proposed algorithm are estimated and some numerical results are provided. The numerical results show that the proposed algorithm is efficient and reliable. [ABSTRACT FROM AUTHOR]

The smoothing-type algorithm has been successfully applied to solve various optimization problems. Based on a new class of smoothing functions, in this paper, we propose a smoothing-type algorithm with a mixed line search for solving the nonlinear complementarity problem. Under suitable conditions, the proposed algorithm is well-defined and maintains global convergence and local superlinear convergence. Preliminary numerical results are also reported and demonstrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper presented an immune particle swarm algorithm for solving generalized Nash equilibrium problem. First it reformulated the generalized Nash equilibrium problem as the nonlinear equations problem by the nonlinear complementarity problem. Then it designed an immune particle swarm algorithm by involving the immune memory function and the antibody concentration inhibition mechanism into the original swarm algorithm. Finally the computer simulation results demonstrate that the proposed algorithm is effective and it not only keeps the variety of the original swarm, but also improves the abilities of seeking the global optimization result and the evolution speed of convergence. [ABSTRACT FROM AUTHOR]

In this paper we propose a class of new large-update primal-dual interior-point algorithms for P( κ) nonlinear complementarity problem (NCP), which are based on a class of kernel functions investigated by Bai et al. in their recent work for linear optimization (LO). The arguments for the algorithms are followed as Peng et al.'s for P( κ) complementarity problem based on the self-regular functions [Peng, J., Roos, C., Terlaky, T.: Self-Regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms, Princeton University Press, Princeton, 2002]. It is worth mentioning that since this class of kernel functions includes a class of non-self-regular functions as special case, so our algorithms are different from Peng et al.'s and the corresponding analysis is simpler than theirs. The ultimate goal of the paper is to show that the algorithms based on these functions have favorable polynomial complexity. [ABSTRACT FROM AUTHOR]

Abstract: Computing traffic equilibria with a general nonadditive route cost disutility function is considered in this paper. Following the user equilibrium (UE) condition, that is, no driver can unilaterally change route to achieve less travel costs, the traffic equilibrium problem (TEP) can be formulated as a nonlinear complementary problem (NCP). In this paper, we propose a semismooth Newton method with a penalized Fischer–Burmeister (PFB) NCP function to solve the NCP formulation of the TEP, and also, we investigate the properties of the proposed method. Numerical results are provided and compared with the classical TEP with additive route cost functions. The results show the algorithm can achieved substantially better performance than the existing approaches. A sensitivity analysis is also conducted to examine the parameter of the proposed nonadditive route cost function. [Copyright &y& Elsevier]

Abstract: The smoothing-type algorithm has been successfully applied to solve various optimization problems. In general, the smoothing-type algorithm is designed based on some monotone line search. However, in order to achieve better numerical results, the non-monotone line search technique has been used in the numerical computations of some smoothing-type algorithms. In this paper, we propose a smoothing-type algorithm for solving the nonlinear complementarity problem with a non-monotone line search. We show that the proposed algorithm is globally and locally superlinearly convergent under suitable assumptions. The preliminary numerical results are also reported. [Copyright &y& Elsevier]

Abstract: Recently, filter methods are extensively studied to handle nonlinear programming problems. Because of good numerical results, filter techniques are attached importance to. The nonlinear complementarity problem can be reformulated as the least -norm solution of an optimization problem. In this paper, basing on the filter technique and the new smoothing function, we present a new Filter-Levenberg–Marquardt method for solving the equation system with the disturbance . Under the assumption that the lever set of the problem is compact, we prove its global convergence. [Copyright &y& Elsevier]

Abstract: In the paper [J.-S. Chen, S. Pan, A family of NCP-functions and a descent method for the nonlinear complementarity problem, Computational Optimization and Applications, 40 (2008) 389–404], the authors proposed a derivative-free descent algorithm for nonlinear complementarity problems (NCPs) by the generalized Fischer–Burmeister merit function: , and observed that the choice of the parameter has a great influence on the numerical performance of the algorithm. In this paper, we analyze the phenomenon theoretically for a derivative-free descent algorithm which is based on a penalized form of and uses a different direction from that of Chen and Pan. More specifically, we show that the algorithm proposed is globally convergent and has a locally -linear convergence rate, and furthermore, its convergence rate will become worse when the parameter decreases. Numerical results are also reported for the test problems from MCPLIB, which further verify the theoretical results obtained. [Copyright &y& Elsevier]

The smoothing-type algorithm has been a powerful tool for solving various optimization problems. In order to improve the numerical results of the algorithm, the nonmonotone line search technique has been used when the algorithm is implemented. However, the theoretical analysis is based on the algorithm with some monotone line search. In this paper, based on the smoothed Kanzow-Kleinmichel NCP function, we propose a smoothing Newton algorithm for solving the nonlinear complementarity problem with a new nonmonotone line search. We show that the nonmonotone algorithm is globally and locally superlinearly convergent under suitable assumptions. The preliminary numerical results are also reported. [ABSTRACT FROM AUTHOR]

Abstract: The nonlinear complementarity problem (denoted by NCP(F)) can be reformulated as the solution of a possibly inconsistent nonsmooth system of equations. Based on the ideas developed in smoothing Newton methods, we approximated the problem of the least l 2-norm solution of the equivalent nonsmooth equations of NCP(F) with a family of parameterized optimization problem with twice continuously differentiable objective functions by making use of a new smoothing function. Then we presented a smoothing Levenberg–Marquardt method to solve the parameterized smooth optimization problem. By using the smooth and semismooth technique, the local quadratic convergence of the proposed method is proved under some suitable assumptions. [Copyright &y& Elsevier]

Abstract: The nonlinear complementarity problem (denoted by NCP(F)) can be reformulated as the least l 2-norm solution of a optimization problem. By introducing a new smoothing function, the problem is approximated by a family of parameterized optimization problems with twice continuously differentiable objective functions. Then a smoothing Levenberg–Marquardt method is applied to solve the parameterized optimization problems. The global convergence of the proposed method is proved under an assumption that the level set of the problem is compact. [Copyright &y& Elsevier]

Abstract: In this paper, we describe H-differential of the implicit Lagrangian function. We show how, under appropriate regularity conditions on an H-differential of f, minimizing the implicit Lagrangian function corresponding to f leads to a solution of the nonlinear complementarity problem. Our results give a unified treatment of such results for C 1-functions and for locally Lipschitzian functions. [Copyright &y& Elsevier]