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Lagrangian Relaxation Method in Approximating the Pareto Front of Multiobjective Optimization Problems

In this paper, we propose that the Lagrangian relaxation approach can be used to approximate the Pareto front of the multiobjective optimization problems. We introduce Lagrangian relaxation approach to solve scalarized subproblems. The scalarization is a technique employed to transform multiple objectives optimization problems into single-objective optimization problems so that existing optimization techniques are used to solve the problems. The relaxation approach exploits transformation and creates a Lagrangian problem in which some of the constraints are replaced from the original problem to make the problem easier to solve.  The method is very effective when the problem is large scale and difficult to solve; this means if the problem has nonconvex and nonsmooth structure, then our proposed method efficiently solves the problem. We succeed in establishing proper Karush Kuhn-Tucker type necessary conditions for our proposed approach. We establish the relation between our proposed approach and the well-known existing approach weighted-sum scalarization methods. We conduct extensive numerical experiments and demonstrated the advantages of the proposed method of adopting a test problem.
GANIT J. Bangladesh Math. Soc. 40.2 (2020) 126-133