Bootstrapping Globally Optimal Variational Calculus Solutions

Abstract

Whereas in a coordinate-dependent setting the Euler-Lagrange equations establish necessary conditions for solving variational problems in which both the integrands of functionals and the resulting paths are assumed to be sufficiently smooth, uniqueness and global optimality are generally hard to prove in the absence of convexity conditions, and often times they may not even exist. This is also true for variational problems on Lie groups, with the Euler-Poincar\'{e} equation establishing necessary conditions. The difficulties compound when integrands and/or the optimal paths are not sufficiently regular, since in this case the classical necessary conditions no longer apply. This article therefore reviews several nonstandard cases where unique globally optimal solutions can be guaranteed, and establishes a ``bootstrapping'' process to build new globally optimal variational solutions on larger spaces from existing ones on smaller spaces. Surprisingly, it is possible to prove global optimality in some nonconvex cases where even the regularity conditions required for classical necessary conditions do not hold. This general theory is then applied to several topics such as optimal framing of curves in three-dimensional Euclidean space, optimal motion interpolation, and optimal reparametrization of video sequences to compare salient actions.