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

40 Pages Posted: 24 Feb 2003

See all articles by Robert M. Freund

## Robert M. Freund

affiliation not provided to SSRN

## Fernando Ordonez

University of Southern California - Epstein Department of Industrial & Systems Engineering

Date Written: February 2003

### Abstract

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.

Keywords: Condition Number, Convex Optimization, Conic Optimization, Duality, Sensitivity Analysis, Perturbation Theory

Suggested Citation

Freund, Robert Michael and Ordonez, Fernando Ivan, On an Extension of Condition Number Theory to Non-Conic Convex Optimization (February 2003). Available at SSRN: https://ssrn.com/abstract=382363 or http://dx.doi.org/10.2139/ssrn.382363