Bilevel programming problems are often reformulated using the Karush-Kuhn-Tucker conditions for the lower level problem resulting in a mathematical program with complementarity constraints (MPCC). First, we present KKT reformulation of the bilevel optimization problems on Riemannian manifolds. Moreover, we show that global optimal solutions of theMPCCcorrespond 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- and C-type optimality conditions for the bilevel problem on Riemannian manifolds are given.

For a better understanding of the bilevel programming on Riemannian manifolds, a semivectorial bilevel programming scheme is proposed in this paper. The semivectorial bilevel programming is firstly transformed into a single-level programming problem by using the Karush–Kuhn–Tucker (KKT) conditions of the lower-level problem, which is convex and satisfies the Slater constraint qualification. Then, the single-level programming is divided into two stages: restoration and minimization, based on which an Inexact Restoration algorithm is developed. Under certain conditions, the stability and convergence of the algorithm are analyzed.

Noticing that $\mathcal{E}$ -convexity, m-convexity and b-invexity have similar structures in their definitions, there are some possibilities to treat these three class of mappings uniformly. For this purpose, the definitions of the $(\mathcal{E},m)$ -convex sets and the b- $(\mathcal{E},m)$ -convex mappings are introduced. The properties concerning operations that preserve the $(\mathcal{E},m)$ -convexity of the proposed mappings are derived. The unconstrained and inequality constrained b- $(\mathcal{E},m)$ -convex programming are considered, where the sufficient conditions of optimality are developed and the uniqueness of the solution to the b- $(\mathcal{E},m)$ -convex programming are investigated. Furthermore, the sufficient optimality conditions and the Fritz–John necessary optimality criteria for nonlinear multi-objective b- $(\mathcal{E},m)$ -convex programming are established. The Wolfe-type symmetric duality theorems under the b- $(\mathcal{E},m)$ -convexity, including weak and strong symmetric duality theorems, are also presented. Finally, we construct two examples in detail to show how the obtained results can be used in b- $(\mathcal{E},m)$ -convex programming.

The bilevel variational inequality on Riemannian manifolds refers to a mathematical problem involving the interaction between two levels of optimization, where one level is constrained by the other level. In this context, we present a variant of Korpelevich’s method specifically designed for solving bilevel variational inequalities on Riemannian manifolds with nonnegative sectional curvature and pseudomonotone vector fields. This variant aims to find a solution that satisfies certain conditions. Through our proposed algorithm, we are able to generate iteration sequences that converge to a solution, given mild conditions. Finally, we provide an example to demonstrate the effectiveness of our algorithm.