On an Extension of Condition Number Theory to Nonconic Convex Optimization
将锥凸优化的条件数现代理论扩展到更一般的非锥格式，其中可行集是任意闭凸集而非锥，并证明相关定理的自然推广。
The purpose of this paper is to extend, as much as possible, the modern theory of condition numbers for conic convex optimization: z * ≔ min x c t x, s.t. Ax−b ∈ C Y , x ∈ C X , to the more general nonconic format: z * ≔ min x c t x, (GP d ) s.t. Ax − b ∈ C Y , x ∈ P, where P is any closed convex set, not necessarily a cone, which we call the ground-set. Although any convex problem can be transformed to conic form, such transformations are neither unique nor natural given the natural description of many problems, thereby diminishing the relevance of data-based condition number theory. Herein we extend the modern theory of condition numbers to the problem format (GP d ). As a byproduct, we are able to state and prove natural extensions of many theorems from the conic-based theory of condition numbers to this broader problem format.