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Nonlinear scalarization in multiobjective optimization with a polyhedral ordering cone

In this work, we consider a multiobjective optimization problem in which the ordering cone is assumed to be polyhedral. In this framework, we characterize proper efficient solutions through nonlinear scalarization and a kind of polyhedral dilating cones. The main results are based on a characterization of weak efficient solutions, for which no convexity hypotheses are required. Moreover, the construction of these dilating cones allows us to obtain scalarization results that are easier to handle, and attractive from a computational point of view, since they are formulated in terms of a perturbation of the matrix that defines the ordering cone. Finally, when the feasible set is given by a cone constraint, we derive necessary and sufficient optimality conditions via a kind of scalar nonlinear Lagrangian.