In the present work we studied a subfield of Applied Mathematics called Riemannian Optimization. The main goal of this subfield is to generalize algorithms, theorems and tools from Mathematical Optimization to the case in which the optimization problem is defined on a Riemannian manifold. As a case study, we implemented some of the main algorithms described in the literature (Gradient Descent, Newton-Raphson and Conjugate Gradient) to solve an optimization problem known as Hartree-Fock. This method is extremely important in the field of Computational Quantum Chemistry and it is a good case study because it is a problem somewhat hard to solve and, as a consequence of this, it requires many tools from Riemannian Optimization. Besides, it is also a good example to see how these algorithms perform in practice.

翻译：本文研究了应用数学的一个子领域——黎曼优化。该子领域的主要目标是将数学优化中的算法、定理及工具推广至优化问题定义在黎曼流形上的情形。作为案例研究，我们实现了文献中描述的一些主要算法（梯度下降法、牛顿-拉夫森法和共轭梯度法），以解决被称为哈特里-福克方法的优化问题。该方法在计算量子化学领域极为重要，且因其求解难度较大，需要借助黎曼优化的诸多工具，因此成为一个良好的研究案例。此外，该案例也有助于观察这些算法在实际中的表现。