- Optimality conditions of the nonlinear programming on Riemannian manifolds Wei Hong Yang (whyangfudan.edu.cn) Leihong Zhang (longzlhgmail.com) Song Ruyi (09210180023fudan.edu.cn) Abstract: 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 $\mathcal{M}$, and discuss the first-order and the second-order optimality conditions. By exploiting the differential geometry structure of the underlying manifold $\mathcal{M},$ we show that, in the language of differential geometry, the first-order and the second-order optimality conditions of the nonlinear programming problem on $\mathcal{M}$ 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 $\mathcal{M}$ is an embedded submanifold of $\mathbb{R}^m$, formed by a set of equality constraints, we show that the optimality conditions can be derived directly from the traditional results on $\mathbb{R}^m$. Keywords: Nonlinear programming， Optimality conditions， Riemannian manifold， Generalized gradient， Hessian Category 1: Nonlinear Optimization Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization ) Category 3: Nonlinear Optimization (Other ) Citation: Submitted. Download: [PDF]Entry Submitted: 08/08/2011Entry Accepted: 08/08/2011Entry Last Modified: 07/16/2012Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.