A Decomposition Augmented Lagrangian Method for Low-rank Semidefinite Programming

Abstract

We develop a decomposition method based on the augmented Lagrangian framework to solve a broad family of semidefinite programming problems possibly with nonlinear objective functions, nonsmooth regularization, and general linear equality/inequality constraints. In particular, the positive semidefinite variable along with a group of linear constraints can be decomposed into a variable on a smooth manifold. The nonsmooth regularization and other general linear constraints are handled by the augmented Lagrangian method. Therefore, each subproblem can be solved by a semismooth Newton method on a manifold. Theoretically, we show that the first and secondorder necessary optimality conditions for the factorized subproblem are also sufficient for the original subproblem under certain conditions. Convergence analysis is established for the Riemannian subproblem and the augmented Lagrangian method. Extensive numerical experiments on large-scale semidefinite programming such as max-cut, nearest correlation estimation, clustering, and sparse principal component analysis demonstrate the strength of our proposed method compared to other state-of-the-art methods.

Solving graph equipartition SDPs on an algebraic variety

“…• We develop an augmented Lagrangian method on an algebraic variety to solve graph equipartition SDP problems and conduct numerical experiments to demonstrate its high efficiency. Although manifold structures are increasingly used in algorithms for solving SDP problems (see [5,21,36]), our novelty here is in developing an algorithm to solve the SDP problems on an algebraic variety that is not necessarily a smooth manifold.…”

“…Thus, the optimal solutions set are the same. Now, we move on to show that problem (36) is related to the well known geometric median problem. This is the key point for solving (36)…”

“…Since any rank-one solution of (36) must have spectral norm equal to √ n, the last inequality implies that rank (R) ≥ 2 and…”

“…By Proposition 3.1, R is smooth. So it is an KKT solution of (36). There exist µ ∈ R r , λ ∈ R n such that the following system holds:…”

“…We use an augmented Lagrangian method on B n,r to solve the problem SDPLR1 in (6). Note that recently the convergence of ALM on manifold has been studied in [21,36,41]. We reformulate problem (6) as follows:…”