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In this paper, based on Chen–Harker–Kanzow–Smale smooth function, we obtain a smoothing homotopy method to solve the second-order cone complementarity problem. The global convergence is ensured under certain non-monotonicity condition for the defined mapping F. The numerical results illustrate that this method is feasible. [ABSTRACT FROM AUTHOR]

In this paper, the system of nonlinear inequalities is considered. The problem is approximated by the parameterized smooth equations which is formed by Chen-Harker-Kanzow-Smale smoothing function. A smoothing homotopy method is proposed to solve the system of smooth equations. For almost all initial points in Euclidean space, the existence and convergence of the homotopy path is proved under certain weaker condition without assumption of the monotonicity. The numerical results illustrate that the method is effective. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new proof for smoothing homotopy method based on the Fischer–Burmeister function to solve the nonlinear complementarity problem under a nonmonotone solution condition. Under this assumption condition imposed on the defined mapping F , global convergence of a smooth curve determined by the referred homotopy equation is established for almost all initial points in R + n and it is actually regarded as an interior point method. Besides, if the initial point is expanded to R n , the global convergence of the homotopy method is ensured under a similar condition. The numerical results are reported and illustrate that the method is efficient for some nonlinear complementarity problems. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a smoothing homotopy method for solving ball-constrained variational inequalities by utilizing a similar Chen–Harker–Kanzow–Smale function to smooth Robinson’s normal equation. Without any monotonicity condition on the defining map , for the starting point chosen almost everywhere in , the existence and convergence of the homotopy pathway are proven. Numerical experiments illustrate that the method is feasible and effective. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new algorithm for tracing the combined homotopy path of the non-convex nonlinear programming problem is proposed. The algorithm is based on the techniques of -cone neighborhood and a combined homotopy interior point method. The residual control criteria, which ensures that the obtained iterative points are interior points, is given by the condition that ensures the -cone neighborhood to be included in the interior part of the feasible region. The global convergence and polynomial complexity are established under some hypotheses. [Copyright &y& Elsevier]

Abstract: In this paper, a new homotopy method for solving the variational inequality problem : find such that , for all , where is a nonempty closed convex subset of and is a continuously differentiable mapping, is proposed. The homotopy equation is constructed based on the smooth approximation to Robinson’s normal equation of variational inequality problem, where the smooth approximation function of the projection function is an arbitrary one such that for any and , . Under a weak condition on the defining mapping , which is needed for the existence of a solution to , for the starting point chosen almost everywhere in , existence and convergence of a smooth homotopy pathway to a solution of are proved. Several numerical experiments indicate that the method is efficient. [Copyright &y& Elsevier]

Abstract: In this paper, using the Gabriel–Moré smoothing function of the median function, a smooth homotopy method for solving nonsmooth equation reformulation of bounded box constrained variational inequality problem VIP() is given. Without any monotonicity condition on the defining map , for starting point chosen almost everywhere in , existence and convergence of the homotopy pathway are proven. Nevertheless, it is also proven that, if the starting point is chosen to be an interior point of the box, the proposed homotopy method can also serve as an interior point method. [Copyright &y& Elsevier]

Abstract: In this paper, based on combined homotopy interior point method we propose an interior point algorithm for convex nonlinear programming. The algorithm ensures that the obtained iterative points are interior points of the feasible set in terms of the technique of β-cone neighborhood. We establish the global convergence of the algorithm. Furthermore, it is shown that the algorithm has iteration complexity. The preliminary numerical experiments indicate that the algorithm is efficient. [Copyright &y& Elsevier]