Hessian Operators on Constraint Manifolds

Abstract

On a constraint manifold we give an explicit formula for the Hessian matrix of a cost function that involves the Hessian matrix of a prolonged function and the Hessian matrices of the constraint functions. We give an explicit formula for the case of the orthogonal group $\mathbf{O}(n)$ by using only Euclidean coordinates on ${\mathbb{R}}^{n^2}$. An optimization problem on $\mathbf{SO}(3)$ is completely carried out. Its applications to nonlinear stability problems are also analyzed.

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