Research On Non-convex Optimization And Non-convex Variational Inequality Problems And Their Algorithms

In mathematical research,optimization problems,variational inequality problem-s are studied mostly in the set is convex.As a new research field,nonconvex problems play an increasingly important role in optimizing theory and variational inequalities.In this paper,we study nonconvex optimization and nonconvex vari-ational inequalities problems.First,we proposed and investigated the algorithm for solving unconstrained nonconvex optimization problem,the objective function of this problem study is true lower semicontinuous,there may be a nonconvex function,under the condition of the Kurdyka-Lojasiewicz property of the objec-tive function,it is proved that the proposed asymptotic algorithm is convergent.In some suitable cases,we also prove that the sequences generated by our algo-rithm have finite length.Meanwhile,we obtain convergence rate result which are related to the function by means of Lojasiewicz exponents.Secondly,the prob-lem of nonconvex variational inequality is studied.The variational inequality has become a kind of excitation and the study appeared in the economic,financial,transportation,and analysis of network structure,source and motive force of most of the problems in elasticity and optimization,it can been see Clarke[22],Ferris[23].For regularized nonconvex mixed variational inequalities,using the auxiliary principle,puts forward the iterative algorithm of the regularized nonconvex mixed variational inequalities,and under the condition of operator is a pseudomonotonic-ity or partially mixed relaxed and strong monotonicity,we prove the convergence of the algorithm.For a system of general nonconvex variational inequalities,using the equivalence between a system of general nonconvex variational inequalities and fixed point problems to build a new perturbation projection iterative algorithm,thus to find the solution of a system of general nonconvex variational inequalities.