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Optimality conditions for the nonlinear programming problems on Riemannian manifolds. (English) Zbl 1322.90096

Summary: 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 a Riemannian 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 \(\mathbb R^m\).

MSC:

90C30 Nonlinear programming
90C46 Optimality conditions and duality in mathematical programming
65K05 Numerical mathematical programming methods
58C05 Real-valued functions on manifolds
49K27 Optimality conditions for problems in abstract spaces
49M37 Numerical methods based on nonlinear programming
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