hajotusmenetelmästä

Hajotusmenetelmä, known in English as the decomposition method, is a general approach used in optimization and computational science to solve complex problems by breaking them down into smaller, more manageable subproblems. This strategy is particularly effective when dealing with large-scale systems where direct solution methods are computationally infeasible or too time-consuming. The core idea is to exploit the structure of the problem, often by separating variables, constraints, or decision stages.

The process typically involves solving these subproblems independently or iteratively, and then combining their solutions to approximate or obtain the optimal solution to the original problem. This iterative refinement is key, as the solution to one subproblem might inform the solution of others, leading to a convergence towards the global optimum. Different variations of the decomposition method exist, each tailored to specific problem structures. Examples include Dantzig-Wolfe decomposition for linear programming and Benders decomposition for mixed-integer programming.

Decomposition methods find applications in various fields, including operations research, engineering, economics, and computer science. They are used in areas such as resource allocation, scheduling, network design, and the analysis of large dynamical systems. The effectiveness of a decomposition method often depends on the degree to which the original problem can be effectively partitioned and the efficiency with which the subproblems can be solved and their solutions aggregated.