Low-rank optimization on the cone of positive semidefinite matrices

We propose an algorithm for solving optimization problems defined on a subset of the cone of symmetric positive semidefinite matrices. This algorithm relies on the factorization X = Y Y T , where the number of columns of Y fixes an upper bound on the rank of the positive semidefinite matrix X. It is thus very effective for solving problems that have a low-rank solution. The factorization X = Y Y T leads to a reformulation of the original problem as an optimization on a particular quotient manifold. The present paper discusses the geometry of that manifold and derives a second-order optimization method with guaranteed quadratic convergence. It furthermore provides some conditions on the rank of the factorization to ensure equivalence with the original problem. In contrast to existing methods, the proposed algorithm converges monotonically to the sought solution. Its numerical efficiency is evaluated on two applications: the maximal cut of a graph and the problem of sparse principal component analysis. © 2010 Society for Industrial and Applied Mathematics.

The effect of carbon encapsulation on the magnetic properties of Ni nanoparticles produced by arc discharge in de-ionized water

Despite intensive research on optimizing the methods for depositing carbon encapsulated ferromagnetic nanoparticles, the effect of the carbon cages remains unclear. In the present work, the effect of the graphitic cages on the magnetization of the ferromagnetic core has been studied by comparing the magnetic properties of pure and carbon encapsulated Ni particles of the same size. The carbon encapsulated Ni particles were formed using an electric arc discharge in de-ionized water between a solid graphite cathode and an anode consisting of Ni and C in a mass ratio of Ni:C = 7:3. This method is shown to have potential for low cost production of carbon encapsulated Ni nanoparticle samples with narrow particle size distributions. X-ray diffraction (XRD) and high resolution transmission electron microscopy (HRTEM) analysis were used to study the crystallography, morphology, and size distribution of the encapsulated and pure Ni nanoparticle samples. The availability of encapsulated particles with various sizes allowed us to elucidate the role of carbon cages in size-dependent properties. Our data suggest that even though encapsulation is beneficial for protection against hostile chemical environments and for avoiding low proximity phenomena, it suppresses the saturation magnetization of the Ni cores.