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.

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Entry Submitted: 08/08/2011
Entry Accepted: 08/08/2011
Entry Last Modified: 07/16/2012

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