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Optimality conditions for the nonlinear programming problems on Riemannian manifolds

Yang Wei Hong (whyang***at***fudan.edu.cn)
Zhang Lei-Hong (longzlh***at***gmail.com)
Song Ruyi (09210180023***at***fudan.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 condition, Riemannian manifold, Generalized gradient, Hessian

Category 1: Nonlinear Optimization (Constrained Nonlinear Optimization )

Category 2: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Citation: Submitted.

Download: [PDF]

Entry Submitted: 07/16/2012
Entry Accepted: 07/16/2012
Entry Last Modified: 11/28/2012

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