A Three-Operator Splitting Perspective of a Three-Block ADMM for Convex Quadratic Semidefinite Programming and Beyond

In recent years, several convergent variants of the multi-block alternating direction method of multipliers (ADMM) have been proposed for solving the convex quadratic semidefinite programming via its dual, which is inherently a [Formula: see text]-block separable convex optimization problem with coupled linear constraints. Among these multi-block ADMM-type algorithms, the modified [Formula: see text]-block ADMM in [Chang, XK, SY Liu and X Li (2016). Modified alternating direction method of multipliers for convex quadratic semidefinite programming. Neurocomputing, 214, 575–586] bears a peculiar feature that the augmented Lagrangian function is not necessarily to be minimized with respect to the block-variable corresponding to the quadratic term in the objective function. In this paper, we lay the theoretical foundation of this phenomenon by interpreting this modified [Formula: see text]-block ADMM as a special implementation of the Davis–Yin [Formula: see text]-operator splitting [Davis, D and WT Yin (2017). A three-operator splitting scheme and its optimization applications. Set-Valued and Variational Analysis, 25, 829–858]. Based on this perspective, we are able to extend this modified [Formula: see text]-block ADMM to a generalized [Formula: see text]-block ADMM, in the sense of [Eckstein, J and DP Bertsekas (1992). On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Mathematical Programming, 55, 293–318], which not only applies to the more general convex composite quadratic programming problems but also admits the flexibility of achieving even better numerical performance.

An SQP-type Method with Superlinear Convergence for Nonlinear Semidefinite Programming

A sequentially semidefinite programming method is proposed for solving nonlinear semidefinite programming problem (NLSDP). Inspired by the sequentially quadratic programming (SQP) method, the algorithm generates a search direction by solving a quadratic semidefinite programming subproblem at each iteration. The [Formula: see text] exact penalty function and a line search strategy are used to determine whether the trial step can be accepted or not. Under mild assumptions, the proposed algorithm is globally convergent. In order to avoid the Maratos effect, we present a modified SQP-type algorithm with the second-order correction step and prove that the fast local superlinear convergence can be obtained under the strict complementarity and the second-order sufficient condition with the sigma term. Finally, some numerical experiments are given to show the effectiveness of the algorithm.

Estimation of Positive Semidefinite Correlation Matrices by Using Convex Quadratic Semidefinite Programming

The correlation matrix is a fundamental statistic that used in many fields. For example, GroupLens, a collaborative filtering system, uses the correlation between users for predictive purposes. Since the correlation is a natural similarity measure between users, the correlation matrix may be used as the Gram matrix in kernel methods. However, the estimated correlation matrix sometimes has a serious defect: although the correlation matrix is originally positive semidefinite, the estimated one may not be positive semidefinite when not all ratings are observed. To obtain a positive semidefinite correlation matrix, the nearest correlation matrix problem has recently been studied in the fields of numerical analysis and optimization. However, statistical properties are not explicitly used in such studies. To obtain a positive semidefinite correlation matrix, we assume an approximate model. By using the model, an estimate is obtained as the optimal point of an optimization problem formulated with information on the variances of the estimated correlation coefficients. The problem is solved by a convex quadratic semidefinite program. A penalized likelihood approach is also examined. The MovieLens data set is used to test our approach.