An Efficient Class of Modulus-Based Matrix Splitting Methods for Nonlinear Complementarity Problems

In this paper, we present an efficient method for finding a numerical solution for nonlinear complementarity problems (NCPs). We first reformulate an NCP as an equivalent system of fixed-point equations and then present a modulus-based matrix splitting iteration method. We prove the convergence of the proposed method with theorems with the relevant conditions. Our preliminary numerical results show that the method is feasible and effective.

SOLOMON: a method for splitting a sample into equivalent subsamples in factor analysis

Nowadays, exploratory and confirmatory factor analyses are two important consecutive steps in an overall analysis process. The overall analysis should start with an exploratory factor analysis that explores the data and establishes a hypothesis for the factor model in the population. Then, the analysis process should be continued with a confirmatory factor analysis to assess whether the hypothesis proposed in the exploratory step is plausible in the population. To carry out the analysis, researchers usually collect a single sample, and then split it into two halves. As no specific splitting methods have been proposed to date in the context of factor analysis, researchers use a random split approach. In this paper we propose a method to split samples into equivalent subsamples similar to one that has already been proposed in the context of multivariate regression analysis. The method was tested in simulation studies and in real datasets.

Efficient Proximal Mapping Computation for Low-Rank Inducing Norms

Low-rank inducing unitarily invariant norms have been introduced to convexify problems with a low-rank/sparsity constraint. The most well-known member of this family is the so-called nuclear norm. To solve optimization problems involving such norms with proximal splitting methods, efficient ways of evaluating the proximal mapping of the low-rank inducing norms are needed. This is known for the nuclear norm, but not for most other members of the low-rank inducing family. This work supplies a framework that reduces the proximal mapping evaluation into a nested binary search, in which each iteration requires the solution of a much simpler problem. The simpler problem can often be solved analytically as demonstrated for the so-called low-rank inducing Frobenius and spectral norms. The framework also allows to compute the proximal mapping of increasing convex functions composed with these norms as well as projections onto their epigraphs.

Steady-State Stand-Alone Power Flow Solvers for Integrated Transmission-Distribution Networks: A Comparison Study and Numerical Assessment

This paper compares and assesses several numerical methods that solve the steady-state power flow problem on integrated transmission-distribution networks. The integrated network model consists of a balanced transmission and an unbalanced distribution network. It is important to analyze these integrated electrical power systems due to the changes related to the energy transition. We classified the existing integration methods as unified and splitting methods. These methods can be applied to homogeneous (complete three-phase) and hybrid (single-phase/three-phase) network models, which results in four approaches in total. These approaches were compared on their accuracy and numerical performance—CPU time and number of iterations—to demonstrate their applicability on large-scale electricity networks. Furthermore, their sensitivity towards the amount of distributed generation and the addition of multiple distribution feeders was investigated. The methods were assessed by running power flow simulations using the Newton–Raphson method on several integrated power systems up to 25,000 unknowns. The assessment showed that unified methods applied to hybrid networks performed the best on these test cases. The splitting methods are advantageous when complete network data sharing between system operators is not allowed. The use of high-performance techniques for larger test cases containing multiple distribution networks will make the difference in speed less significant.