Điều kiện tối ưu cho các dạng nghiệm hữu hiệu của bài toán tối ưu vector
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Abstract
In the paper, optimality conditions are established for various types of efficient solutions associated with an arbitrary set and its recession cone in the vector optimization problem. First, the separation theorems for convex sets, the concepts of improvement sets, and the recession cone of an arbitrary set are recalled, and some of their new properties are also investigated. Then, the model of the vector optimization problem is examined, along with its Pareto efficient solutions corresponding to an arbitrary set and Benson efficient solutions corresponding to the recession cone. Finally, by employing the linear scalarization method, necessary and sufficient optimality conditions for these efficient solutions are derived.
Tóm tắt
Trong bài báo, việc thiết lập các điều kiện tối ưu cho các dạng nghiệm hữu hiệu liên quan đến tập hợp bất kỳ và nón lùi xa của nó cho bài toán tối ưu vector đã được thực hiện. Đầu tiên, các định lý tách cho tập lồi, các khái niệm về tập cải tiến và nón lùi xa của một tập bất kỳ được nhắc lại, đồng thời một số tính chất mới của chúng cũng được khảo sát. Sau đó, mô hình bài toán tối ưu vector cùng với nghiệm hữu hiệu Pareto tương ứng với tập hợp bất kỳ và nghiệm hữu hiệu Benson tương ứng với nón lùi xa của bài toán tối ưu vector được xem xét. Cuối cùng, bằng cách sử dụng phương pháp vô hướng hóa tuyến tính, các điều kiện cần và đủ tối ưu cho các nghiệm hữu hiệu này được thiết lập.
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