Abstract
We are concerned with a nonsmooth multiobjective optimization problem with inequality constraints. In order to obtain our main results, we give the definitions of the generalized convex functions based on the generalized directional derivative. Under the above generalized convexity assumptions, sufficient and necessary conditions for optimality are given without the need of a constraint qualification. Then we formulate the dual problem corresponding to the primal problem, and some duality results are obtained without a constraint qualification.
Highlights
A multiobjective problem is a problem where two or more objective functions are to be minimized on an implicitly constrained feasible set. In such a problem for optimality conditions and duality results, we often deal with constraint qualifications
A constraint qualification assumes some regularity of the constraint functions near the optimal solution, in particular to exclude a cusp on the boundary of the feasible region
In some approaches to multiobjective optimization problems, the necessary conditions for efficiency are derived under the same constraint qualifications as in nonlinear programming with a scalar-valued objective function
Summary
A multiobjective problem is a problem where two or more objective functions are to be minimized on an implicitly constrained feasible set. In some approaches to multiobjective optimization problems, the necessary conditions for efficiency are derived under the same constraint qualifications as in nonlinear programming with a scalar-valued objective function. Weir and Mond [8] defined dual problems for the scalar-valued programming problems where the usual convexity requirement for duality was relaxed and a constraint qualification was not needed. 3. Necessary and sufficient optimality conditions Consider the following nonlinear programming problem: min f0(x) s.t. gi(x) ≤ 0, i ∈ P, x ∈ C, where f0 and gi, i ∈ P, are scalar, locally Lipschitz functions and C is a convex set. To prove the results we need the following theorems for the nonlinear program (SP)
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