In applied mathematics and computer science, variable splitting is a decomposition method that relaxes a set of constraints. [1]
When the variable x appears in two sets of constraints, it is possible to substitute the new variables x1 in the first constraints and x2 in the second, and then join the two variables with a new "linking" constraint, [2] which requires that
This new linking constraint can be relaxed with a Lagrange multiplier; in many applications, a Lagrange multiplier can be interpreted as the price of equality between x1 and x2 in the new constraint.
For many problems, when the equality of the split variables is relaxed, then the system is decomposed, and each subsystem can be solved independently, at substantial reduction of computing time and memory storage. A solution to the relaxed problem (with variable splitting) provides an approximate solution to the original problem: further, the approximate solution to the relaxed problem provides a "warm start", a good initialization of an iterative method for solving the original problem (having only the x variable).
This was first introduced by Kurt O. Jörnsten, Mikael Näsberg, Per A. Smeds in 1985. At the same time, M. Guignard and S. Kim introduced the same idea under the name Lagrangean Decomposition (their papers appeared in 1987). The original references are (1) Variable Splitting: A New Lagrangean Relaxation Approach to Some Mathematical Programming Models Authors Kurt O. Jörnsten, Mikael Näsberg, Per A. Smeds Volumes 84-85 of LiTH MAT R.: Matematiska Institutionen Publisher - University of Linköping, Department of Mathematics, 1985 Length - 52 pages; and (2) Lagrangean Decomposition: A Model Yielding Stronger Bounds, Authors Monique Guignard and Siwhan Kim, Mathematical Programming, 39(2), 1987, pp. 215-228. [2] [3] [4]
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics, numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.
Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks to optimize a multivariate quadratic function subject to linear constraints on the variables. Quadratic programming is a type of nonlinear programming.
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In mathematical optimization, Dantzig's simplex algorithm is a popular algorithm for linear programming.
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Multi-disciplinary design optimization (MDO) is a field of engineering that uses optimization methods to solve design problems incorporating a number of disciplines. It is also known as multidisciplinary system design optimization (MSDO), and Multidisciplinary Design Analysis and Optimization (MDAO).
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Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematics. It is a subfield of numerical analysis, and a type of linear algebra. Computers use floating-point arithmetic and cannot exactly represent irrational data, so when a computer algorithm is applied to a matrix of data, it can sometimes increase the difference between a number stored in the computer and the true number that it is an approximation of. Numerical linear algebra uses properties of vectors and matrices to develop computer algorithms that minimize the error introduced by the computer, and is also concerned with ensuring that the algorithm is as efficient as possible.
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