Abstract:In this paper, a method for scalarizing optimization problems whose final space is endowed with a binary relation is stated without assuming any additional hypothesis on the data of the problem. By this approach, nondominated and minimal solutions are characterized in terms of solutions of scalar optimization problems whose objective functions are the post-composition of the original objective with scalar functions satisfying suitable properties. The obtained results generalize some recent ones stated in quasi… Show more
“…We retrieve here the notions of (strictly) -preserving and (strictly) -representing scalarizing functions (considered e.g. in [13] and more recently in [12]) and we refer them to the quasi order relation . We note that the order preserving properties of a function are closely related to monotonicity, while the order representing properties compare level sets of the function with the order structure of the underlying space.…”
The robust optimization approach can be used to tackle uncertain vector problems by considering worst case scenarios. In this context, notions of robust efficient solutions which are coherent with a set-valued minimization process have been introduced in literature in order to avoid unduly pessimistic attitudes (see e.g. Ehrgott et al. in Eur. J. Oper. Res. 239(1), 17–31, 2014). We address the question whether scalarization and robustification can be commuted in a non componentwise framework. We prove that the commutation of the two approaches is ensured under appropriate assumptions. To this purpose, we identify a class of scalarization processes that ensure necessary and sufficient robust optimality conditions through the direct scalarization of the uncertain vector optimization problem, without explicitly passing through the set-valued formulation of the problem.
“…We retrieve here the notions of (strictly) -preserving and (strictly) -representing scalarizing functions (considered e.g. in [13] and more recently in [12]) and we refer them to the quasi order relation . We note that the order preserving properties of a function are closely related to monotonicity, while the order representing properties compare level sets of the function with the order structure of the underlying space.…”
The robust optimization approach can be used to tackle uncertain vector problems by considering worst case scenarios. In this context, notions of robust efficient solutions which are coherent with a set-valued minimization process have been introduced in literature in order to avoid unduly pessimistic attitudes (see e.g. Ehrgott et al. in Eur. J. Oper. Res. 239(1), 17–31, 2014). We address the question whether scalarization and robustification can be commuted in a non componentwise framework. We prove that the commutation of the two approaches is ensured under appropriate assumptions. To this purpose, we identify a class of scalarization processes that ensure necessary and sufficient robust optimality conditions through the direct scalarization of the uncertain vector optimization problem, without explicitly passing through the set-valued formulation of the problem.
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