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On an Extension of Condition Number Theory to Non-Conic Convex Optimization

The purpose of this paper is to extend, as much as possible, the modern theory of condition numbers for conic convex optimization: z_* = min cx subject to Ax-b \in C_Y , x \in C_X, to the more general non-conic format: (GP_d) z_* = min cx subject to Ax-b \in C_Y , x \in P, where P is any closed convex set, not necessarily a cone, which we call the ground-set. While the conic format has been essential to recent theoretical developments in convex optimization theory (particularly interior-point methods) and any convex problem can be transformed to conic form, such transformations are neither unique nor natural given the natural description and data for 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 state and prove natural extensions of many theorems from the conic-based theory of condition numbers to this broader problem format.