Optimality conditions for the nonlinear programming problems on Riemannian manifolds

In recent years, many traditional optimization methods have been successfully generalized to minimize objective functions on manifolds. In this paper, we first extend the general traditional constrained optimization problem to a nonlinear programming problem built upon a general Riemannian manifold M , and discuss the first-order and the second-order optimality conditions. By exploiting the differential geometry structure of the underlying manifold M , we show that, in the language of differential geometry, the first-order and the second-order optimality conditions of the nonlinear programming problem on M coincide with the traditional optimality conditions. When the objective function is nonsmooth Lipschitz continuous, we extend the Clarke generalized gradient, tangent and normal cone, and establish the first-order optimality conditions. For the case when M is an embedded submanifold of Rm, formed by a set of equality constraints, we show that the optimality conditions can be derived directly from the traditional results on Rm.