Multi-objective Variational Curves
研究了黎曼流形上曲线速度与加速度的L2范数线性组合的临界点，即张力黎曼三次曲线，将其视为多目标优化问题，并构造了球面和环面上的帕累托前沿，发现环面上的前沿不连通且揭示了具有相同边界数据的两种不同黎曼三次曲线。
Abstract Riemannian cubics in tension are critical points of the linear combination of two objective functionals, namely the squared $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> norms of the velocity and acceleration of a curve on a Riemannian manifold. We view this variational problem of finding a curve as a multi-objective optimization problem and construct the Pareto fronts for some given instances where the manifold is a sphere and where the manifold is a torus. The Pareto front for the curves on the torus turns out to be particularly interesting: the front is disconnected and it reveals two distinct Riemannian cubics with the same boundary data, which is the first known nontrivial instance of this kind. We also discuss some convexity conditions involving the Pareto fronts for curves on general Riemannian manifolds.