Modified interior-point method for large-and-sparse low-rank semidefinite programs

Abstract

Semidefinite programs (SDPs) are powerful theoretical tools that have been studied for over two decades, but their practical use remains limited due to computational difficulties in solving large-scale, realistic-sized problems. In this paper, we describe a modified interior-point method for the efficient solution of large-and-sparse low-rank SDPs, which finds applications in graph theory, approximation theory, control theory, sum-of-squares, etc. Given that the problem data is large-and-sparse, conjugate gradients (CG) can be used to avoid forming, storing, and factoring the large and fully-dense interior-point Hessian matrix, but the resulting convergence rate is usually slow due to ill-conditioning. Our central insight is that, for a rank-k, size-n SDP, the Hessian matrix is ill-conditioned only due to a rank-nk perturbation, which can be explicitly computed using a size-n eigendecomposition. We construct a preconditioner to "correct" the low-rank perturbation, thereby allowing preconditioned CG to solve the Hessian equation in a few tens of iterations. This modification is incorporated within SeDuMi, and used to reduce the solution time and memory requirements of large-scale matrix-completion problems by several orders of magnitude.Assumption 1 (Nondegeneracy). We assume:1) (Slater's condition) There exist X 0, y, and S 0, such that A i • X = b i and i y i A i + S = C. 2) (Strict complementarity) rank (X ) + rank (S ) = n.These are generic properties of SDPs, and are satisfied by almost all instances [10]. Note that Slater's condition is satisfied in solvers like SeDuMi [11] and MOSEK [12] using the homogenous self-dual embedding technique [13].We further assume that the data matrices A 1 , . . . , A m are structured in a way that allow certain matrix-implicit operations to be efficiently performed.Assumption 2 (Sparsity). Define the matrix A [vec A 1 , . . . , vec A m ]. We assume that matrix-vector products with A, A T and (A T A) −1 may each be applied in O(m) flops and memory.

Conic Optimization Theory: Convexification Techniques and Numerical Algorithms

“…The following iteration bound is standard; a proof can be found in standard references like [172] and [179]. A preconditioner that guarantees joint condition number κ = O(1) was described in [30]. The preconditioner based on the insight that the scaling matrix W can be decomposed into two components,…”

“…However, its use required almost as much time and memory as a single iteration of the regular interior-point method. The modified interior-point method of [30], which we had described in this subsection, makes the same idea efficient by utilizing the Sherman-Morrison-Woodbury formula.…”

Barrier and penalty methods for low-rank semidefinite programming with application to truss topology design

“…3. Using one of the second-order algorithms with Cholesky factorization replaced by an iterative method for the solution of the linear systems Hx = g; see, e.g., [21,37,42]. The last approach is particularly attractive whenever n m, and it addresses both bottlenecks of a second-order SDP solver:…”

“…Our preconditioner is motivated by the work of Zhang and Lavaei [42]. They present a preconditioner for the CG method within a standard IP method that makes it more efficient for large-and-sparse low-rank SDPs.…”

Semidefinite Programming via Generalized Proximal Point Algorithm

“…This approach has enabled parallel implementations that can solve larger instances with the use of supercomputers [68]. In a series of works, Zhang and Lavaei have presented SDP algorithms that can properly take advantage of the problem's sparsity [69,70]. Primal-dual operator splitting for SDP…”