# 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_x {c'x | Ax-b \in C_Y, x \in C_X }, to the more general non-conic format: (GP_d): z_* := min_x {c'x | 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. 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.

## Citation

Working paper #2003-01, USC-Department of Indstrial and Systems Engineering, Feb/2003