Duality in Multi-Objective Optimization Under Uncertainty (2014)
In this paper we extend to a multi-objective optimization with a interval-valued function and real valued constraints, the concepts of Wolfes duality elaborated on [1] and [2], for the interval-valued mono-objective case with real constraints or intervals-valued. First of all, being supported on the Wolfes duality theory valued set [3], we perform an extension to optimize a deterministic multi-objective function, and with real valued constraints of a proposal made by Wolfe [4] for a duality in a optimization with objective function and real valued constraints. The theorems 3.1 and 3.3 are theorems of duality in a weak and strong sense, respectively; i.e, they guarantee that all objective value of a dual problem are lesser than all the objective value of the primal problem, and, under some conditions, the furthest values of the primal and dual problem are the same. Secondly, being supported on [3] and, in the Wolfes duality for a deterministic multi-objective case, we developed Wolfes duality concepts for an optimization under uncertainty with a interval-valued function criteria with real valued constraints. Lem-mas 4.3, 4.4 and proposition 4.5 constitute a duality in a weak sense, and theorems 4.8 and 4.11 constitute a duality in a strong sense.