微分流形上的最优化算法

Optimization algorithms on differentiable manifolds

指出了流形算法中利用测地线寻找最优解存在附加度量结构和计算复杂的问题,根据流形的局部与欧氏空间零点的开邻域光滑同胚这一性质,利用坐标变换把非线性等式约束优化问题转化为无约束优化问题,利用坐标变换而不是黎曼几何结构给出了函数取得极值的充分和必要条件,构造了一种映射梯度算法,并证明这种算法是线性收敛的.

The problem of the descent algorithm along geodesic on Riemannian manifolds is provided. A technique for the optimization algorithm for the differentiable function on differentiable manifolds is given. The constrained optimization problem is converted to the unconstrained case with the special choices of coordinate transformation and the optimality condition for the constrained optimization problem is given. Moreover, a mapping gradient method is developed and linear convergence of the mapping gradient method is established.