Unifying Efficiency and Weak Efficiency in Generalized Quasiconvex Vector Minimization on the Real-line
MetadataShow full item record
Usually the concepts of efficient and weakly efficient solution are associated to a multiobjective optimization problem. Both notions may be described in terms of a preference relation determined by a closed convex cone with nonempty interior - typically the nonnegative orthant of a finite dimensional space. We present a unified approach for dealing with both notions at the same time under generalized quasi convexity assumptions on the objectives defined on the realline. Since most algorithms in scalar minimization involve the solvability of a one-dimensional problem to find the next iterate, it is expected that our results be applied in the vector case. We established several characterizations of the nonemptiness of the solution set, and also various characterizations when besides boundedness is required. To that end we used a notion of relaxed convexity for vector functions introduced ealier by one of the authors.