# Minimizing Condition Number via Convex Programming

@article{Lu2011MinimizingCN,
title={Minimizing Condition Number via Convex Programming},
author={Zhaosong Lu and Ting Kei Pong},
journal={SIAM J. Matrix Anal. Appl.},
year={2011},
volume={32},
pages={1193-1211}
}
• Published 1 November 2011
• Computer Science, Mathematics
• SIAM J. Matrix Anal. Appl.
In this paper we consider minimizing the spectral condition number of a positive semidefinite matrix over a nonempty closed convex set $\Omega$. We show that it can be solved as a convex programming problem, and moreover, the optimal value of the latter problem is achievable. As a consequence, when $\Omega$ is positive semidefinite representable, it can be cast into a semidefinite programming problem. We then propose a first-order method to solve the convex programming problem. The…

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## References

SHOWING 1-10 OF 24 REFERENCES
Optimizing Condition Numbers
• Mathematics
SIAM J. Optim.
• 2009
The condition number is a Clarke regular strongly pseudoconvex function and it is proved that a global solution of the problem can be approximated by an exact or an inexact solution of a nonsmooth convex program.
Primal-dual first-order methods with O (1/e) iteration-complexity for cone programming.
• Computer Science, Mathematics
• 2011
First-order methods suitable for solving primal-dual convex (smooth and/or nonsmooth) minimization reformulations of the cone programming problem are discussed, and a variant of Nesterov's optimal method is proposed which has outperformed the latter one in computational experiments.
Primal-dual first-order methods with $${\mathcal {O}(1/\epsilon)}$$ iteration-complexity for cone programming
• Computer Science, Mathematics
Math. Program.
• 2011
This paper discusses first-order methods suitable for solving primal-dual convex and nonsmooth minimization reformulations of the cone programming problem, and proposes a variant of Nesterov’s optimal method which has outperformed the latter one in the authors' computational experiments.
A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization
• Computer Science
Math. Program.
• 2003
A nonlinear programming algorithm for solving semidefinite programs (SDPs) in standard form that replaces the symmetric, positive semideFinite variable X with a rectangular variable R according to the factorization X=RRT.
Probing the Pareto Frontier for Basis Pursuit Solutions
• Mathematics
SIAM J. Sci. Comput.
• 2008
A root-finding algorithm for finding arbitrary points on a curve that traces the optimal trade-off between the least-squares fit and the one-norm of the solution is described, and it is proved that this curve is convex and continuously differentiable over all points of interest.
Solving semidefinite-quadratic-linear programs using SDPT3
• Computer Science
Math. Program.
• 2003
Computational experiments with linear optimization problems involving semidefinite, quadratic, and linear cone constraints (SQLPs) are discussed and computational results on problems from the SDPLIB and DIMACS Challenge collections are reported.
On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators
• Computer Science, Mathematics
Math. Program.
• 1992
This paper shows, by means of an operator called asplitting operator, that the Douglas—Rachford splitting method for finding a zero of the sum of two monotone operators is a special case of the proximal point algorithm, which allows the unification and generalization of a variety of convex programming algorithms.
On the Closedness of the Linear Image of a Closed Convex Cone
Very simple and intuitive necessary conditions are presented that unify, and generalize seemingly disparate, classical sufficientconditions such as polyhedrality of the cone, and Slater-type conditions.
Convex Optimization
• Computer Science
IEEE Transactions on Automatic Control
• 2006
A comprehensive introduction to the subject of convex optimization shows in detail how such problems can be solved numerically with great efficiency.
A Matlab toolbox for optimization over symmetric cones
• Computer Science
• 1999
This paper describes how to work with SeDuMi, an add-on for MATLAB, which lets you solve optimization problems with linear, quadratic and semidefiniteness constraints by exploiting sparsity.