Block Coordinate Descent Methods for Semidefinite Programming

Zaiwen Wen (), Donald Goldfarb () and Katya Scheinberg ()
Additional contact information
Zaiwen Wen: Shanghai Jiaotong University
Donald Goldfarb: Columbia University
Katya Scheinberg: Lehigh University

Chapter Chapter 19 in Handbook on Semidefinite, Conic and Polynomial Optimization, 2012, pp 533-564 from Springer

Abstract: Abstract We consider in this chapter block coordinate descent (BCD) methods for solving semidefinite programming (SDP) problems. These methods are based on sequentially minimizing the SDP problem’s objective function over blocks of variables corresponding to the elements of a single row (and column) of the positive semidefinite matrix X; hence, we will also refer to these methods as row-by-row (RBR) methods. Using properties of the (generalized) Schur complement with respect to the remaining fixed (n − 1)-dimensional principal submatrix of X, the positive semidefiniteness constraint on X reduces to a simple second-order cone constraint. It is well known that without certain safeguards, BCD methods cannot be guaranteed to converge in the presence of general constraints. Hence, to handle linear equality constraints, the methods that we describe here use an augmented Lagrangian approach. Since BCD methods are first-order methods, they are likely to work well only if each subproblem minimization can be performed very efficiently. Fortunately, this is the case for several important SDP problems, including the maxcut SDP relaxation and the minimum nuclear norm matrix completion problem, since closed-form solutions for the BCD subproblems that arise in these cases are available. We also describe how BCD can be applied to solve the sparse inverse covariance estimation problem by considering a dual formulation of this problem. The BCD approach is further generalized by using a rank-two update so that the coordinates can be changed in more than one row and column at each iteration. Finally, numerical results on the maxcut SDP relaxation and matrix completion problems are presented to demonstrate the robustness and efficiency of the BCD approach, especially if only moderately accurate solutions are desired.

Keywords: Cholesky Factorization; Augmented Lagrangian Method; Matrix Completion; Augmented Lagrangian Function; Coordinate Descent Method (search for similar items in EconPapers)
Date: 2012
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:isochp:978-1-4614-0769-0_19

Ordering information: This item can be ordered from
http://www.springer.com/9781461407690

DOI: 10.1007/978-1-4614-0769-0_19

Access Statistics for this chapter

More chapters in International Series in Operations Research & Management Science from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().