Lagrangian decomposition

It has been also demonstrated that a Lagrangian decomposition can significantly reduce the computational burden associated with the solution of monolithic problems. In this chapter, the Optimal Condition Decomposition was applied. This technique facilitates the process required to iteratively update the Lagrangian multipliers. 

The local spin values obtained in a decomposition scheme may either be local expectation values as evident from the last section. Since we are interested merely in Slater determinants describing local high-spin centers, it is most suitable to employ local (Sza) values as properties to be constrained in such a multiplier scheme. If we recall that the (Sza) expectation values should be equal to the ideal Msa and denote ideal values as M al values, the Lagrangian can be chosen to be (135,136)... 

There are two major approaches to elaboration of computationally efficient algorithms. These are based on Lagrangian relaxation and Bender s decomposition. A short overview of these methods is provided here. Readers are referred to Avriel and Golany (1996) for a detailed coverage of mathematical programming. 

The Optimal Condition Decomposition (OCD), which is a particular case of the Lagrangian relaxation procedure, is applied to overcome the computational cost of solving the monolithic problem, which integrates SC design-retrofitting and product development pipeline management formulations. Further details about this decomposition strategy can be found in Appendix A.7.1. 

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