Linganisha mbinu

	Njia ya Lagrangian Iliyoimarishwa	Uharibifu wa Benders	Kuzalisha nguzo (Dantzig-Wolfe)
Nyanja	Utafiti wa Operesheni	Utafiti wa Operesheni	Utafiti wa Operesheni
Familia	Machine learning	Machine learning	Machine learning
Mwaka wa asili	1969	1962	1960
Mwanzilishi	Magnus R. Hestenes and M. J. D. Powell	Jacques F. Benders	George B. Dantzig and Philip Wolfe
Aina	algorithm	algorithm	algorithm
Chanzo asilia	Hestenes, M. R. (1969). Multiplier and gradient methods. Journal of Optimization Theory and Applications, 4(5), 303-320.	Benders, J. F. (1962). Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik, 4(1), 238-252.	Dantzig, G. B., & Wolfe, P. (1960). Decomposition principle for linear programs. Operations Research, 8(1), 101-111.
Majina mbadala	method of multipliers, augmented Lagrangian, ADMM	cutting plane method, constraint generation	Dantzig-Wolfe decomposition, column generation method
Zinazohusiana	3	3	3
Muhtasari	The Augmented Lagrangian Method, developed by Magnus R. Hestenes and M. J. D. Powell in 1969, is a powerful technique for solving constrained optimization problems. It converts a constrained problem into a sequence of unconstrained subproblems by augmenting the Lagrangian with a quadratic penalty term, enabling efficient solution of large-scale problems including convex and nonconvex cases.	Benders Decomposition, introduced by Jacques F. Benders in 1962, is a powerful algorithmic framework for solving large-scale mixed-integer programming (MIP) problems. It decomposes the problem into a master problem (controlling complicating variables) and subproblems (handling remaining variables), using cutting planes generated from subproblem dual information to iteratively tighten the master problem.	Column Generation, developed by George B. Dantzig and Philip Wolfe in 1960, is a powerful optimization technique for solving large-scale linear programming problems with special structure. Also known as Dantzig-Wolfe Decomposition, it decomposes the problem into a master problem (restricted to a subset of variables/columns) and a pricing subproblem (identifying new variables), iteratively improving the solution by introducing only relevant columns.