A new monotonicity, "Equation missing"<!-- image only, no MathML or LaTex -->-monotonicity, is introduced, and the resolvant operator of an "Equation missing"<!-- image only, no MathML or LaTex -->-monotone operator is proved to be single-valued and Lipschitz continuous. With the help of the resolvant operator, the positively semidefinite general variational inequality (VI) problem VI "Equation missing"<!-- image only, no MathML or LaTex --> is transformed into a fixed point problem of a nonexpansive mapping. And a proximal point algorithm is constructed to solve the fixed point problem, which is proved to have a global convergence under the condition that "Equation missing"<!-- image only, no MathML or LaTex --> in the VI problem is strongly monotone and Lipschitz continuous. Furthermore, a convergent path Newton method is given for calculating "Equation missing"<!-- image only, no MathML or LaTex -->-solutions to the sequence of fixed point problems, enabling the proximal point algorithm to be implementable.

In this paper, we study the optimality conditions and perform a stability analysis for the second-order cone constrained variational inequalities (SOCCVI) problem. The Lagrange function and Karush–Kuhn–Tucker (KKT) condition of the SOCCVI problem is given, and the optimality conditions for the SOCCVI problem are studied. Then, the second-order sufficient condition satisfying the constrained nondegenerate condition is proved. The strong second-order sufficient condition is defined. And the nonsingularity of Clarke’s generalized Jacobian of the KKT point, the strong regularity of the KKT point, the uniform second-order growth condition, the strong stability of the KKT point, and the local Lipschtiz homeomorphism of the KKT point for the SOCCVI problem are proved to be equivalent to each other. Then, the stability theorem of the SOCCVI problem is obtained.

In this paper, we smooth the Karush-Kuhn-Tucker (KKT) functions for the classical variational inequality problem, then it is transformed to an unconstrained optimization problem. We firstly establish the second-order differential equation system involving two time-dependent parameters for solving the unconstrained optimization problem. The global convergence theorem for the second-order differential equation system is proved. At last, four numerical experiments are reported to verify the effectiveness of the second-order differential equation method for solving the classical variational inequality problem.

This paper focuses on solving a class of equilibrium problems, namely, the fixed point problem of set-valued mappings with second-order cone double constraints. Under certain conditions, the variational inequality form of the fixed point problem of set-valued mappings with second-order cone double constraints is obtained by using the generalized saddle point theory three times. The alternating direction method is used to solve the fixed point problem of set-valued mappings with second-order cone double constraints, and the global convergence of the algorithm is proved. Finally, numerical results of solving five examples with an inexact alternating direction method are given, and the feasibility and effectiveness of the algorithm are demonstrated by comparing with other algorithms.