I encountered some optimization problem on the special Euclidean group SE(3) at work and wonder how to solve it. The current approach of my colleagues was to use a local parametrization of the manifold and then applying a function from a standard optimization toolbox. However I am also aware of the recent developments on optimization problems on manifolds (see e.g. <http://www.eeci-institute.eu/GSC2011/Photos-EECI/EECI-GSC-2011-M5/book_AMS.pdf>). This way there is no parametrization needed. Is there any other advantage? Have the two approached been compared somewhere?



How to solve optimization problems on manifolds?
To complement Christian Clason's comment: there is usually no need to compute geodesics to optimize over manifolds directly. The usual replacement used for optimization purposes is called a retraction. A retraction is an approximation of the exponential map (which generates geodesics), accurate up to first order. For various manifolds of practical interest, computationally inexpensive retractions are known. For example, for the sphere, retracting the tangent vector $u$ at $x$ is often done as $R\_x(u) = \frac{x+u}{\|x+u\|\_2}$. For $u$ small, this agrees with the exponential of $u$ at $x$ up to first order.


Edit: in addition to Absil et al.'s excellent book, you may also find my introduction to optimization on manifolds helpful:
<http://www.nicolasboumal.net/book>.


