The condition number of a differentiable convex function, namely the ratio of its smoothness to strong convexity constants, is closely tied to fundamental properties of the function. In particular, the condition number of a quadratic convex function is the square of the aspect ratio of a canonical ellipsoid associated to the function. Furthermore, the condition number of a function bounds the linear rate of convergence of the gradient descent algorithm for unconstrained convex minimization. We propose a condition number of a differentiable convex function relative to a reference convex set and distance function pair. This relative condition number is defined as the ratio of relative smoothness to relative strong convexity constants. We show that the relative condition number extends the main properties of the traditional condition number both in terms of its geometric insight and in terms of its role in characterizing the linear convergence of first-order methods for constrained convex minimization. When the reference set X is a convex cone or a polyhedron and the function f is of the form $$f = g\circ A$$ f = g ∘ A , we provide characterizations of and bounds on the condition number of f relative to X in terms of the usual condition number of g and a suitable condition number of the pair ( A , X ).

中文翻译：

---

函数相对于集合的条件数

可微凸函数的条件数，即其平滑度与强凸性常数的比率，与函数的基本性质密切相关。特别地，二次凸函数的条件数是与该函数相关联的标准椭球的纵横比的平方。此外，函数的条件数限制了用于无约束凸最小化的梯度下降算法的线性收敛速度。我们提出了相对于参考凸集和距离函数对的可微凸函数的条件数。这个相对条件数被定义为相对平滑度与相对强凸度常数的比率。我们表明，相对条件数扩展了传统条件数的主要属性，无论是在几何洞察力方面，还是在表征约束凸最小化的一阶方法的线性收敛方面的作用方面。当参考集 X 是凸锥或多面体且函数 f 的形式为 $$f = g\circ A$$ f = g ∘ A 时，我们提供了 f 的条件数的特征和边界相对于X 表示 g 的通常条件数和对 ( A , X ) 的合适条件数。