The nearest correlation matrix problem is to find a correlation matrix which is closest to a given symmetric matrix in the Frobenius norm. The well studied dual approach is to reformulate this problem as an unconstrained continuously differentiable convex optimization problem. Gradient methods and quasi-Newton methods like BFGS have been used directly to obtain globally convergent methods. Since the objective function in the dual approach is not twice continuously differentiable, these methods converge at best linearly. In this paper, we investigate a Newton-type method for the nearest correlation matrix problem. Based on recent developments on strongly semismooth matrix valued functions, we prove the quadratic convergence of the proposed Newton method. Numerical experiments confirm the fast convergence and the high efficiency of the method.

For any function f from R to R, one can define a corresponding function on the space of n × n (block-diagonal) real symmetric matrices by applying f to the eigenvalues of the spectral decomposition. We show that this matrix-valued function inherits from f the properties of continuity, (local) Lipschitz continuity, directional differentiability, Fréchet differentiability, continuous differentiability, as well as (ρ-order) semismoothness. Our analysis uses results from nonsmooth analysis as well as perturbation theory for the spectral decomposition of symmetric matrices. We also apply our results to the semidefinite complementarity problem, addressing some basic issues in the analysis of smoothing/semismooth Newton methods for solving this problem.

Abstract. The Nearest Euclidean distance matrix problem (NEDM) is a fundamental computational problem in applications such as multidimensional scaling and molecular conformation from nuclear magnetic resonance data in computational chemistry. Especially in the latter application, the problem is often large scale with the number of atoms ranging from a few hundreds to a few thousands. In this paper, we introduce a semismooth Newton method that solves the dual problem of (NEDM). We prove that the method is quadratically convergent. We then present an application of the Newton method to NEDM with H-weights. We demonstrate the superior performance of the Newton method over existing methods including the latest quadratic semi-definite programming solver. This research also opens a new avenue towards efficient solution methods for the molecular embedding problem.

Euclidean distance embedding appears in many high-profile applications including wireless sensor network localization, where not all pairwise distances among sensors are known or accurate. The classical Multi-Dimensional Scaling (cMDS) generally works well when the partial or contaminated Euclidean Distance Matrix (EDM) is close to the true EDM, but otherwise performs poorly. A natural step preceding cMDS would be to calculate the nearest EDM to the known matrix. A crucial condition on the desired nearest EDM is for it to have a low embedding dimension and this makes the problem nonconvex. There exists a large body of publications that deal with this problem. Some try to solve the problem directly and some are the type of convex relaxations of it. In this paper, we propose a numerical method that aims to solve this problem directly. Our method is strongly motivated by the majorized penalty method of Gao and Sun for low-rank positive semi-definite matrix optimization problems. The basic geometric object in our study is the set of EDMs having a low embedding dimension. We establish a zero duality gap result between the problem and its Lagrangian dual problem, which also motivates the majorization approach adopted. Numerical results show that the method works well for the Euclidean embedding of Network coordinate systems and for a class of problems in large scale sensor network localization and molecular conformation.

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Higham (2002, IMA J. Numer. Anal., 22, 329-343) considered two types of nearest correlation matrix problems, namely the W -weighted case and the H -weighted case. While the W -weighted case has since been well studied to make several Lagrangian dual-based efficient numerical methods available, the H -weighted case remains numerically challenging. The difficulty of extending those methods from the W -weighted case to the H -weighted case lies in the fact that an analytic formula for the metric projection onto the positive semidefinite cone under the H -weight, unlike the case under the W -weight, is not available. In this paper we introduce an augmented Lagrangian dual-based approach that avoids the explicit computation of the metric projection under the H -weight. This method solves a sequence of unconstrained convex optimization problems, each of which can be efficiently solved by an inexact semismooth Newton method combined with the conjugate gradient method. Numerical experiments demonstrate that the augmented Lagrangian dual approach is not only fast but also robust.

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Abstract-Kruskal's stress minimization, though nonconvex and nonsmooth, has been a major computational model for dissimilarity data in multidimensional scaling. Semidefinite Programming (SDP) relaxation (by dropping the rank constraint) would lead to a high number of SDP cone constraints. This has rendered the SDP approach computationally challenging even for problems of small size. In this paper, we reformulate the stress optimization as an Euclidean Distance Matrix (EDM) optimization with box constraints. A key element in our approach is the conditional positive semidefinite cone with rank cut. Although nonconvex, this geometric object allows a fast computation of the projection onto it and it naturally leads to a majorization-minimization algorithm with the minimization step having a closed-form solution. Moreover, we prove that our EDM optimization follows a continuously differentiable path, which greatly facilitated the analysis of the convergence to a stationary point. The superior performance of the proposed algorithm is demonstrated against some of the state-of-the-art solvers in the field of sensor network localization and molecular conformation.