The Weinstein conjecture predicts that there is at least one periodic orbit for every compact contact hypersurface in a symplectic manifold, which turns out an important research direction in symplectic topology and contact topology. According to the methods of the research, this paper surveys the study on the Weinstein conjecture using the variational method and the pseudo-holomorphic curves method.
Let
In this paper, by constructing a new smoothing complementary function, we reformulate the nonlinear complementarity problem as a nonlinear smooth system of equations. Combining non-monotonic line search techniques with an inexact Broyden-like algorithm, we establish a nonmonotone inexact Broyden-like algorithm. The global and local quadratic convergence of this method is proved under suitable conditions. Numerical experiments show that the algorithm is effective for solving nonlinear complementarity problems.
In this paper, through the use of image space analysis, optimality conditions for a class of variational inequalities with cone constraints are proposed. By virtue of the nonlinear scalarization function, known as the Gerstewitz function, three nonlinear weak separation functions, two nonlinear regular weak separation functions and a nonlinear strong separation function are introduced. According to nonlinear separation functions, some optimality conditions of the weak and strong alternative for variational inequalities with cone constraints are derived.
