Theoretical And Algorithmic Studies On Bilevel Programming Problems And Bilevel Variational Inequalities On Riemannian Manifolds

Bilevel programming problems are often reformulated using the Karush-KuhnTucker conditions for the lower level problem resulting in a mathematical program with complementarity constraints(MPCC).The extension of the optimization problem to Riemannian manifolds also has many advantages.Taking Euclidean space as a special Riemannian manifold,the constrained optimization problem in Euclidean space can be regarded as an unconstrained optimization problem on a special Riemannian manifold.Optimization on Riemannian manifolds can also transform non-convex optimization problems on Euclidean Spaces into convex optimization problems on Riemannian manifolds.Based on this theory,the following work is done in this paper.First,we present KKT reformulation of the bilevel optimization problems on Riemannian manifolds.Moreover,we show that global optimal solutions of the MPCC correspond to global optimal solutions of the bilevel problem on the Riemannian manifolds provided the lower level convex problem satisfies the Slater’s constraint qualification.But the relationship between the local solutions of the bilevel problem and its corresponding MPCC is incomplete equivalent.We then also show by examples that these correspondences can fail if the Slater’s constraint qualification fails to hold at lower-level convex problem.In addition,M-type and C-type optimality conditions for the bilevel problem on Riemannian manifolds are given.Then we introduce a bilevel variational inequality on Riemannian manifolds and give the definition of pseudomonotone vector fields on Riemannian manifolds.A Korpelevich algorithm for solving bilevel variational inequalities on Riemannian manifolds with non-negative sectional curvature and pseudo-monotone vector fields is given by using the projection method on manifolds.In order to obtain the convergence of the algorithm,we propose some mild conditions,prove that the iterative sequence generated by the algorithm is convergent,and give an example to explain the effectiveness of the algorithm.Finally,in order to further study the bilevel programming problem on Riemannian manifolds,we propose a semi-vector bilevel programming problem.The semi-vector bilevel programming problem is transformed into a single-level programming problem by using KKT conditions of the lower level problem,which is convex and satisfies the Slater constraint qualification.On this basis,we divide the single-level programming problem into two stages and give an Inexact-Restoration Algorithm.Under certain conditions,the stability and convergence of the algorithm are analyzed.