论文标题
部分可观测时空混沌系统的无模型预测
Newton and interior-point methods for (constrained) nonconvex-nonconcave minmax optimization with stability and instability guarantees
论文作者
论文摘要
我们解决了使用牛顿类型方法（包括内点方法）找到非Convex-Nonconcave MinMax优化的局部解决方案的问题。我们修改了这些方法的Hessian矩阵，以便在每个步骤中，修改后的牛顿更新方向可以看作是局部近似Minmax问题的二次程序的解决方案。此外，我们表明，通过以适当的方式选择修改，算法迭代的唯一稳定平衡点是局部Minmax点。结果，如果该点是局部minmax，则算法只能趋向于平衡点，如果该点不是局部minmax，它将逃脱。使用数值示例，我们表明算法的计算时间与Hessian中的非零元素的数量大致线性缩放。
We address the problem of finding a local solution to a nonconvex-nonconcave minmax optimization using Newton type methods, including interior-point ones. We modify the Hessian matrix of these methods such that, at each step, the modified Newton update direction can be seen as the solution to a quadratic program that locally approximates the minmax problem. Moreover, we show that by selecting the modification in an appropriate way, the only stable equilibrium points of the algorithm's iterations are local minmax points. As a consequence, the algorithm can only converge towards an equilibrium point if such point is a local minmax, and it will escape if the point is not a local minmax. Using numerical examples, we show that the computation time of our algorithm scales roughly linearly with the number of nonzero elements in the Hessian.
