Parallel and distributed computation: numerical methods
Parallel and distributed computation: numerical methods
Mathematical Programming: Series A and B
Nonmonotone Spectral Projected Gradient Methods on Convex Sets
SIAM Journal on Optimization
Preconditioning techniques for large linear systems: a survey
Journal of Computational Physics
Convex Optimization
On the Closedness of the Linear Image of a Closed Convex Cone
Mathematics of Operations Research
Probing the Pareto Frontier for Basis Pursuit Solutions
SIAM Journal on Scientific Computing
SIAM Journal on Optimization
Mathematical Programming: Series A and B
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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 computational results show that our method is usually faster than the standard interior point solver SeDuMi [J. F. Sturm, Optim. Methods Softw., 11/12 (1999), pp. 625-653] while producing a comparable solution. We also study a closely related problem, that is, finding an optimal preconditioner for a positive definite matrix. This problem is not convex in general. We propose a convex relaxation for finding positive definite preconditioners. This relaxation turns out to be exact when finding optimal diagonal preconditioners.