Bibliographic Information: Günther, C., Khazayel, B., Strugariu, R., & Tammer, C. (2024). Bishop-Phelps Type Scalarization for Vector Optimization in Real Topological-Linear Spaces. arXiv preprint arXiv:2410.10026v1.
Research Objective: This paper aims to extend the existing conic scalarization method for vector optimization problems from real normed spaces to the more general setting of real topological-linear spaces. This is achieved by employing Bishop-Phelps type seminorm-linear functions instead of norm-linear functions.
Methodology: The authors utilize the concept of Bishop-Phelps type cones and functions to develop a new scalarization method. They explore the properties of these cones and functions, particularly their cone representation properties. The paper also delves into nonlinear cone separation techniques using Bishop-Phelps type separating cones and functions. These techniques are then applied to derive scalarization results for vector optimization problems in real topological-linear spaces.
Key Findings: The research establishes new results on cone separation in real topological-linear spaces using Bishop-Phelps type separating cones and seminorm-linear functions. It demonstrates that Bishop-Phelps type scalarization can effectively generate weakly, properly efficient solutions for vector optimization problems in this general setting. The paper provides new scalarization theorems for various efficiency concepts, including weak efficiency, proper efficiency with respect to a cone-valued map, and Henig proper efficiency.
Main Conclusions: The authors successfully extend Kasimbeyli's conic scalarization method to real topological-linear spaces, broadening the applicability of this approach. The use of Bishop-Phelps type seminorm-linear functions proves to be effective in this more general context. The paper provides a theoretical foundation for solving vector optimization problems in a wider range of applications, including optimization under uncertainty and problems in mathematical finance.
Significance: This research significantly contributes to the field of vector optimization by extending a well-established scalarization method to a more general setting. This opens up new possibilities for solving complex optimization problems in various fields.
Limitations and Future Research: The paper focuses on the theoretical foundations of this extended scalarization method. Future research could explore practical implementations and applications of this method to specific problem domains. Additionally, investigating the computational efficiency of this approach would be beneficial.
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