In this work we propose a method to perform optimization on manifolds. We assume to have an objective function $f$ defined on a manifold and think of it as the potential energy of a mechanical system. By adding a momentum-dependent kinetic energy we define its Hamiltonian function, which allows us to write the corresponding Hamiltonian system. We make it conformal by introducing a dissipation term: the result is the continuous model of our scheme. We solve it via splitting methods (Lie-Trotter and leapfrog): we combine the RATTLE scheme, approximating the conserved flow, with the exact dissipated flow. The result is a conformal symplectic method for constant stepsizes. We also propose an adaptive stepsize version of it. We test it on an example, the minimization of a function defined on a sphere, and compare it with the usual gradient descent method.

翻译：本文提出了一种在流形上进行优化的方法。我们假设目标函数$f$定义在流形上，并将其视为力学系统的势能。通过引入依赖于动量的动能，我们定义了其哈密顿函数，从而能够写出相应的哈密顿系统。通过引入耗散项使其成为共形系统：这构成了我们方案的连续模型。我们采用分裂方法（李-特罗特法和蛙跳法）对其进行求解：将逼近守恒流的RATTLE方案与精确耗散流相结合。结果得到了恒定步长的共形辛方法。我们还提出了其自适应步长版本。我们通过一个示例——球面上函数的最小化问题——对该方法进行了测试，并将其与常规梯度下降法进行了比较。