Convexity Analysis of Optimization Framework of Attitude Determination from Vector Observations
Provides theoretical justification for the absence of local optima in a common aerospace engineering problem, but the finding is incremental as it confirms convexity under standard practice.
The paper proves that the loss function for attitude determination from vector observations is convex under quaternion normalization, eliminating local optima issues in derivative-based optimization algorithms.
In the past several years, there have been several representative attitude determination methods developed using derivative-based optimization algorithms. Optimization techniques e.g. gradient-descent algorithm (GDA), Gauss-Newton algorithm (GNA), Levenberg-Marquadt algorithm (LMA) suffer from local optimum in real engineering practices. A brief discussion on the convexity of this problem is presented recently \cite{Ahmed2012} stating that the problem is neither convex nor concave. In this paper, we give analytic proofs on this problem. The results reveal that the target loss function is convex in the common practice of quaternion normalization, which leads to non-existence of local optimum.