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We investigate the problem of optimal transport in the so-called Kantorovich form, i.e. given two Radon measures on two compact sets, we seek an optimal transport plan which is another Radon measure on the product of the sets that has these two measures as marginals and minimizes a certain cost function. We consider quadratic regularization of the problem, which forces the optimal transport plan to be a square integrable function rather than a Radon measure. We derive the dual problem and show strong duality and existence of primal and dual solutions to the regularized problem. Then we derive two algorithms to solve the dual problem of the regularized problem: A Gauss–Seidel method and a semismooth quasi-Newton method and investigate both methods numerically. Our experiments show that the methods perform well even for small regularization parameters. Quadratic regularization is of interest since the resulting optimal transport plans are sparse, i.e. they have a small support (which is not the case for the often used entropic regularization where the optimal transport plan always has full measure).

Motivated by recent literature demonstrating the surprising effectiveness of the heuristic application of progressive hedging (PH) to stochastic mixed-integer programming (SMIP) problems, we provide theoretical support for the inclusion of integer variables, bridging the gap between theory and practice. We provide greater insight into the following observed phenomena of PH as applied to SMIP where optimal or at least feasible convergence is observed. We provide an analysis of a modified PH algorithm from a different viewpoint, drawing on the interleaving of (split) proximal-point methods (including PH), Gauss–Seidel methods, and the utilisation of variational analysis tools. Through this analysis, we show that under mild conditions, convergence to a feasible solution should be expected. In terms of convergence analysis, we provide two main contributions. First, we contribute insight into the convergence of proximal-point-like methods in the presence of integer variables via the introduction of the notion of persistent local minima. Secondly, we contribute an enhanced Gauss–Seidel convergence analysis that accommodates the variation of the objective function under mild assumptions. We provide a practical implementation of a modified PH and demonstrate its convergent behaviour with computational experiments in line with the provided analysis.

We contribute improvements to a Lagrangian dual solution approach applied to large-scale optimization problems whose objective functions are convex, continuously differentiable and possibly nonlinear, while the non-relaxed constraint set is compact but not necessarily convex. Such problems arise, for example, in the split-variable deterministic reformulation of stochastic mixed-integer optimization problems. We adapt the augmented Lagrangian method framework to address the presence of nonconvexity in the non-relaxed constraint set and to enable efficient parallelization. The development of our approach is most naturally compared with the development of proximal bundle methods and especially with their use of serious step conditions. However, deviations from these developments allow for an improvement in efficiency with which parallelization can be utilized. Pivotal in our modification to the augmented Lagrangian method is an integration of the simplicial decomposition method and the nonlinear block Gauss–Seidel method. An adaptation of a serious step condition associated with proximal bundle methods allows for the approximation tolerance to be automatically adjusted. Under mild conditions optimal dual convergence is proven, and we report computational results on test instances from the stochastic optimization literature. We demonstrate improvement in parallel speedup over a baseline parallel approach.

The propagation of soliton waves is simulated through splices in quadratic optical media, in which fluctuations of dielectric parameters occur. A new numerical scheme was developed to solve the complex system of partial differential equations (PDE) that describes the problem. Our numerical approach to solve the complex problem was based on the mathematical theory of Taylor series of complex functions. In this context, we adapted the Finite Difference Method (FDM) to approximate derivatives of complex functions and resolve the algebraic system, which results from the discretization, implicitly, by means of the relaxation Gauss-Seidel method. The mathematical modeling of local fluctuations of dielectric properties of optical media was performed by Gaussian functions. By simulating soliton wave propagation in optical fibers with Gaussian fluctuations in their dielectric properties, it was observed that the perturbed soliton numerical solution presented higher sensitivity to fluctuations in the dielectric parameter β, a measure of the nonlinearity intensity in the fiber. In order to verify whether the fluctuations of β parameter in the splices of the optical media generate unstable solitons, the propagation of a soliton wave, subject to this perturbation, was simulated for large time intervals. Considering various geometric configurations and intensities of the fluctuations of parameter β, it was found that the perturbed soliton wave stabilizes, i.e., the amplitude of the wave oscillations decreases as the values of propagation distance increases. Therefore, the propagation of perturbed soliton wave presents numerical stability when subjected to local Gaussian fluctuations (perturbations) of the dielectric parameters of the optical media.

This paper concerns the generalized Nash equilibrium problem of polynomials (GNEPP). We apply the Gauss–Seidel method and Moment-SOS relaxations to solve GNEPPs. The convergence of the Gauss–Seidel method is known for some special GNEPPs, such as generalized potential games (GPGs). We give a sufficient condition for GPGs and propose a numerical certificate, based on Putinar’s Positivstellensatz. Numerical examples for both convex and nonconvex GNEPPs are given for demonstrating the efficiency of the proposed method.

Penetration of distributed generators (DGs) to the grid is transcending because of the importance given to green energy. Microgrids are gaining attention because of DGs and local control to reduce peak demand on the grid. Power flow analysis in microgrids must be considered while expanding the microgrids. Even though the conventional methods for power flow analysis apply to grid-connected mode, they cannot be used for an islanded mode of microgrid operation. Many modifications to the existing approach were proposed in the literature such as no slack bus, variable system frequency, and droop-controlled generators. A low-voltage microgrid of short distance which is for small communities is considered in the work. For such transmission lines, the resistance will be more than the line reactance. This paper focuses on modifying the conventional Gauss–Seidel method for the power flow analysis in low-voltage short transmission islanded microgrid. The power flow equations are modified considering there is no slack bus, and DG models are formulated for low-voltage, short transmission networks with droop control. The effectiveness of these considerations is illustrated by conducting simulation studies on a six-bus network and its effect on system frequency, real, and reactive power losses are also analyzed. The results obtained are compared with the existing results in the latest literature in which optimization techniques were used for droop coefficients calculation, and it is found that in the proposed approach, the real power losses are reduced by 2 kW without using any optimization algorithm for calculating the droop coefficients. Hence, the proposed approach is a good choice for power flow analysis in low-voltage microgrids for smaller communities.

By utilizing some elements of each row of the majorization matrix associated with the coefficient tensor, we propose a preconditioner, and present the corresponding preconditioned Gauss–Seidel method for solving M -tensor multi-linear system. Theoretically, we give the convergence and comparison theorems of the proposed preconditioned Gauss–Seidel method. Numerically, we show the correctness of theoretical results and the efficiency of the proposed preconditioner by some examples.

This paper presents a greedy randomized average block sampling Gauss–Seidel (GRABGS) method for solving sparse linear least-squares problems. The GRABGS method utilizes a novel probability criterion to collect the control index set of coordinates, and minimizes the quadratic convex objective by performing multiple accurate line searches on average per iteration. The probability criterion aims to capture subvectors whose norms are relatively large. Additionly, the GRABGS method is categorized as a member of randomized block Gauss–Seidel methods, which can be employed for parallel implementations. The convergence analysis encompasses two types of extrapolation stepsizes: constant and adaptive. It is proved that the GRABGS method converges to the unique solution of the sparse linear least-squares problem when the matrix has full column rank. Numerical examples demonstrate the superiority of this method over the greedy randomized coordinate descent method and several existing state-of-the-art block Gauss–Seidel methods.

By combining the preconditioner I + S α by Li et al. (Appl Numer Math 134:105–121, 2018) and some elements of the last row of the majorization matrix associated with the coefficient tensor, we propose a new preconditioner and present the corresponding preconditioned Gauss–Seidel method for solving multi-linear systems with M -tensors. Theoretically, we give the convergence and comparison theorems of the proposed preconditioned Gauss–Seidel method. Numerical examples are given to show our theoretical results and the efficiency of the proposed preconditioner.

Purpose The purpose of this research is to study the dynamic behavior of hydrostatic squeeze film dampers made of four hydrostatic pads, fed through four capillary restrictors with micropolar lubricant. Design/methodology/approach The modified version of Reynolds equation is solved numerically by the finite differences and the Gauss–Seidel methods to determine the pressure field generated on the hydrostatic bearing flat pads. In the first step, the effects of the pad dimension ratios on the stiffness and damping coefficients are investigated. In the second step, the damping factor is evaluated with respect to the micropolar properties. Findings The analysis revealed that the hydrostatic squeeze film dampers lubricated with micropolar lubricants produces the maximum damping factor for characteristic length of micropolar lubricant less than 5, while the same bearing operating with Newtonian lubricants reaches its maximum damping factor at eccentricity ratios larger than 0.4. Originality/value The results obtained show that the effects of micropolar lubricants on the dynamic performances are predominantly affected by the pad geometry and eccentricity ratio.