Decomposition-Coordination Methods by Augmented Lagrangian: Applications

1984 ◽  
pp. 166-194
Author(s):  
Roland Glowinski
Keyword(s):  
Augmented Lagrangian ◽  
Coordination Methods

Exploiting Soft Constraints within Decomposition and Coordination Methods for Sub-hourly Unit Commitment

2020 ◽  
Author(s):  
Niranjan Raghunathan ◽  
Mikhail Bragin ◽  
Bing Yan ◽  
Peter Luh ◽  
Khosrow Moslehi ◽  
...  
Keyword(s):  
Unit Commitment ◽  
Daily Basis ◽  
Time Limit ◽  
Soft Constraints ◽  
Problem Complexity ◽  
Surrogate Optimization ◽  
Parallel Version ◽  
Improved Performance ◽  
Coordination Methods ◽  
Relaxed Problem

Unit commitment (UC) is an important problem solved on a daily basis within a strict time limit. While hourly UC problems are currently considered, they may not be flexible enough with the fast-changing demand and the increased penetration of intermittent renewables. Sub-hourly UC is therefore recommended. This, however, will significantly increase problem complexity even under the deterministic setting, and current methods may not be able to obtain good solutions within the time limit. In this paper, deterministic sub-hourly UC is considered, with the innovative exploitation of soft constraints – constraints that do not need to be strictly satisfied, but with predetermined penalty coefficients for their violations. The key idea is the “surrogate optimization” concept that ensures multiplier convergence within “surrogate” Lagrangian relaxation as long as the “surrogate optimality condition” is satisfied without the need to optimally solve the “relaxed problem.” Consequently, subproblems can still be formed and optimized when soft constraints are not relaxed, leading to a drastically reduced number of multipliers and improved performance. To further enhance the method, a parallel version is developed. Testing results on the Polish system demonstrate the effectiveness and robustness of both the sequential and parallel versions at finding high-quality solutions within the time limit.

The characteristics of transforming coordinates from the geocentric system WGS-84 for the Mercator projection under low latitudes conditions

2018 ◽  
Vol 940 (10) ◽  
pp. 2-6
Author(s):  
J.A. Younes ◽  
M.G. Mustafin
Keyword(s):  
Coordinate System ◽  
Satellite System ◽  
Programming Algorithm ◽  
Low Latitude ◽  
Model Calculations ◽  
Mercator Projection ◽  
Low Latitudes ◽  
Rectangular Coordinates ◽  
Global Navigation Satellite ◽  
Coordination Methods

The issue of calculating the plane rectangular coordinates using the data obtained by the satellite observations during the creation of the geodetic networks is discussed in the article. The peculiarity of these works is in conversion of the coordinates into the Mercator projection, while the plane coordinate system on the base of Gauss-Kruger projection is used in Russia. When using the technology of global navigation satellite system, this task is relevant for any point (area) of the Earth due to a fundamentally different approach in determining the coordinates. The fact is that satellite determinations are much more precise than the ground coordination methods (triangulation and others). In addition, the conversion to the zonal coordinate system is associated with errors; the value at present can prove to be completely critical. The expediency of using the Mercator projection in the topographic and geodetic works production at low latitudes is shown numerically on the basis of model calculations. To convert the coordinates from the geocentric system with the Mercator projection, a programming algorithm which is widely used in Russia was chosen. For its application under low-latitude conditions, the modification of known formulas to be used in Saudi Arabia is implemented.

scholarly journals Augmented Lagrangian preconditioners for the Oseen–Frank model of nematic and cholesteric liquid crystals

2021 ◽  
Author(s):  
Jingmin Xia ◽  
Patrick E. Farrell ◽  
Florian Wechsung
Keyword(s):  
Liquid Crystals ◽  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Mass Matrix ◽  
Unit Length ◽  
Mesh Size ◽  
Schur Complement ◽  
Cholesteric Liquid Crystals ◽  
Multigrid Algorithm ◽  
The Cost

AbstractWe propose a robust and efficient augmented Lagrangian-type preconditioner for solving linearizations of the Oseen–Frank model arising in nematic and cholesteric liquid crystals. By applying the augmented Lagrangian method, the Schur complement of the director block can be better approximated by the weighted mass matrix of the Lagrange multiplier, at the cost of making the augmented director block harder to solve. In order to solve the augmented director block, we develop a robust multigrid algorithm which includes an additive Schwarz relaxation that captures a pointwise version of the kernel of the semi-definite term. Furthermore, we prove that the augmented Lagrangian term improves the discrete enforcement of the unit-length constraint. Numerical experiments verify the efficiency of the algorithm and its robustness with respect to problem-related parameters (Frank constants and cholesteric pitch) and the mesh size.

scholarly journals On a Robust and Efficient Numerical Scheme for the Simulation of Stationary 3-Component Systems with Non-Negative Species-Concentration with an Application to the Cu Deposition from a Cu-(β-alanine)-Electrolyte

Algorithms ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 113
Author(s):  
Stephan Daniel Schwoebel ◽  
Thomas Mehner ◽  
Thomas Lampke
Keyword(s):  
Augmented Lagrangian ◽  
Computation Time ◽  
Species Distributions ◽  
Chemical Processes ◽  
Diffusion Reaction ◽  
Augmented Lagrangian Methods ◽  
Major Question ◽  
Component Systems ◽  
Reaction Models ◽  
Reaction Equations

Three-component systems of diffusion–reaction equations play a central role in the modelling and simulation of chemical processes in engineering, electro-chemistry, physical chemistry, biology, population dynamics, etc. A major question in the simulation of three-component systems is how to guarantee non-negative species distributions in the model and how to calculate them effectively. Current numerical methods to enforce non-negative species distributions tend to be cost-intensive in terms of computation time and they are not robust for big rate constants of the considered reaction. In this article, a method, as a combination of homotopy methods, modern augmented Lagrangian methods, and adaptive FEMs is outlined to obtain a robust and efficient method to simulate diffusion–reaction models with non-negative concentrations. Although in this paper the convergence analysis is not described rigorously, multiple numerical examples as well as an application to elctro-deposition from an aqueous Cu2+-(β-alanine) electrolyte are presented.

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