In the setting of normed spaces ordered by a convex cone, we define a set-valued optimization problem where the feasible set is given by a cone-constraint and we derive Lagrange optimality conditions for weakly efficient solutions of the problem following both the set and the vector solution criterion. For the set criterion, a lower set less preorder is considered. The optimality conditions are expressed in terms of coderivatives and are obtained through scalarization, by using a nonlinear scalarizing functional based on the oriented distance.
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The authors are grateful to the anonymous referees for their useful suggestions and remarks.
This work was partially supported by the Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI) (Spain) and Fondo Europeo de Desarrollo Regional (FEDER) under project PGC2018-096899-B-I00 (MCIU/AEI/FEDER, UE).
Dedicated to Marco A. López on the occasion of his 70th birthday.
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Huerga, L., Jiménez, B. & Novo, V. Lagrange Multipliers in Convex Set Optimization with the Set and Vector Criteria. Vietnam J. Math. 48, 345–362 (2020). https://doi.org/10.1007/s10013-020-00404-4
- Set optimization
- Lagrange multipliers
- Lower set preorder relation
- Scalarization in set optimization
- Oriented distance
Mathematics Subject Classification (2010)