This paper begins with a brief review of affine invariance and its significance for iterative algorithms. It then explores the existence of affine invariant descent directions for unconstrained minimization. While there may exist several affine invariant descent directions for smooth functions at a given point, it is shown that for quadratic functions, there exists exactly one invariant descent direction in the strictly convex case and generally none in the case where the Hessian is singular or indefinite. These results can be generalized to smooth nonlinear functions and have implications regarding the initialization of minimization algorithms. They stand in contrast to recent works on constrained convex and nonconvex optimization for which there may exist an affine invariant ‘frame’ that depends on the feasible set and that can be used to define an affine invariant descent direction.

本文首先简要介绍了仿射不变性及其对迭代算法的重要性。然后，它探讨了无约束最小化的仿射不变下降方向的存在。虽然在给定点上可能存在多个仿射不变的下降方向，但对于二次函数，它表明在严格凸情况下，仅存在一个不变的下降方向，而在Hessian是奇异或不确定的情况下通常不存在。这些结果可以推广为平滑非线性函数，并且对最小化算法的初始化有影响。