Working Paper On an Extension of Condition Number Theory to Non-Conic Convex Optimization
Working Paper On an Extension of Condition Number Theory to Non-Conic Convex Optimization
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关于条件数论到非圆锥凸优化的扩展的工作论文
DOI：
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发表时间：
2016
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通讯作者：
Fernando Ord
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作者：
Fernando Ord
本文的目的是尽可能地推广锥凸优化的条件数的现代理论：z*：= minzctxs. t。Ax B Cy C Cx，更一般的非圆锥格式：z*：= minx ctx（GPd）s.t. Ax-b E Cy X P，其中P是任何闭凸集，不一定是锥，我们称之为基集。虽然任何凸问题都可以转换成圆锥形式，但这种转换既不唯一也不自然，因为许多问题的自然描述，从而减少了基于数据的条件数论的相关性。在这里，我们将条件数的现代理论扩展到问题格式（GPd）。作为副产品，我们能够陈述和证明许多定理的自然扩展，从基于圆锥曲线的条件数理论到这个更广泛的问题格式。
The purpose of this paper is to extend, as much as possible, the modern theory of condition numbers for conic convex optimization: z* := minz ctx s.t. Ax b Cy C Cx , to the more general non-conic format: z* := minx ctx (GPd) s.t. Ax-b E Cy X P, where P is any closed convex set, not necessarily a cone, which we call the groundset. 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 (GPd). 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.