Bishop-Phelps Type Scalarization for Vector Optimization in Real Topological-Linear Spaces

Abstract

It is well-known that scalarization techniques (e.g., in the sense of Gerstewitz; Kasimbeyli; Pascoletti and Serafini; Zaffaroni) are useful for generating (weakly, properly) efficient solutions of vector optimization problems. One recognized approach is the conic scalarization method in vector optimization in real normed spaces proposed by Kasimbeyli (2010, SIAM J Optim 20), which is based on augmented dual cones and Bishop-Phelps type (norm-linear) scalarizing functions. In this paper, we present new results on cone separation in real topological-linear spaces by using Bishop-Phelps type separating cones / separating seminorm-linear functions. Moreover, we study some extensions of known scalarization results in vector optimization (in the sense of Eichfelder; Gerstewitz; Jahn; Kasimbeyli; Pascoletti and Serafini). On this basis, we propose a Bishop-Phelps type scalarization method for vector optimization problems in real topological-linear spaces, which is based on Bishop-Phelps type cone-representing and cone-monotone scalarizing functions (e.g., Gerstewitz scalarizing functions or seminorm-linear scalarizing functions). Thus, our method also extends Kasimbeyli's conic scalarization method from real normed spaces to real topological-linear spaces. Within this framework, we derive new Bishop-Phelps type scalarization results for the concepts of weak efficiency and different types of proper efficiency.