On an Extension of Condition Number Theory to Non-Conic Convex Optimization
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Author(s)
Freund, Robert
Ordonez, Fernando
Date Issued
February 26, 2003
Series/Report no.
MIT Sloan School of Management Working Paper;4286-03
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
Subjects
Condition Number
Convex Optimization
Conic Optimization
Duality
Sensitivity Analysis
Perturbation Theory
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