论文标题
解散riemannian优化的约束
Dissolving Constraints for Riemannian Optimization
论文作者
论文摘要
在本文中，我们考虑了$ \ mathbb {r}^n $的封闭嵌入式子手机的优化问题，这些submanifolds由约束$ c（x）= 0 $定义。我们为这些Riemannian优化问题提出了一类约束溶解方法。在这些提出的方法中，解决riemannian优化问题被转移到了名为CDF的约束溶解函数的无约束最小化中。与现有的确切惩罚功能不同，CDF的确切梯度和Hessian易于计算。我们研究了CDF的理论特性，并证明原始问题和CDF具有相同的一阶和二阶固定点，局部最小化器和可行区域附近的lojasiewicz指数。值得注意的是，我们所提出的约束溶解方法的收敛性能可以直接从现有的丰富结果中遗传为无约束的优化。因此，所提出的约束溶解方法从不受约束的优化到Riemannian优化增加了捷径。几个说明性的例子进一步证明了我们提出的约束溶解方法的潜力。
In this paper, we consider optimization problems over closed embedded submanifolds of $\mathbb{R}^n$, which are defined by the constraints $c(x) = 0$. We propose a class of constraint dissolving approaches for these Riemannian optimization problems. In these proposed approaches, solving a Riemannian optimization problem is transferred into the unconstrained minimization of a constraint dissolving function named CDF. Different from existing exact penalty functions, the exact gradient and Hessian of CDF are easy to compute. We study the theoretical properties of CDF and prove that the original problem and CDF have the same first-order and second-order stationary points, local minimizers, and Łojasiewicz exponents in a neighborhood of the feasible region. Remarkably, the convergence properties of our proposed constraint dissolving approaches can be directly inherited from the existing rich results in unconstrained optimization. Therefore, the proposed constraint dissolving approaches build up short cuts from unconstrained optimization to Riemannian optimization. Several illustrative examples further demonstrate the potential of our proposed constraint dissolving approaches.
