Existence of efficient and properly efficient solutions to problems of constrained vector optimization

The paper is devoted to the existence of global optimal solutions for a general class of nonsmooth problems of constrained vector optimization without boundedness assumptions on constraint set. The main attention is paid to the two major notions of optimality in vector problems: Pareto efficiency and proper efficiency in the sense of Geoffrion. Employing adequate tools of variational analysis and generalized differentiation, we first establish relationships between the notions of properness, M-tameness, and the Palais–Smale conditions formulated for the restriction of the vector cost mapping on the constraint set. These results are instrumental to derive verifiable necessary and sufficient conditions for the existence of Pareto efficient solutions in vector optimization. Furthermore, the developed approach allows us to obtain new sufficient conditions for the existence of Geoffrion-properly efficient solutions to such constrained vector problems. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.

Kim D.S.1 , Mordukhovich B.S. 2, 3 , Phạm T.-S.4 , Van Tuyen N.
  • 1 Department of Applied Mathematics, Pukyong National University, Busan, 48513, South Korea
  • 2 Department of Mathematics, Wayne State University, Detroit, MI 48202, United States
  • 3 RUDN University, Moscow, 117198, Russian Federation
  • 4 Department of Mathematics, University of Dalat, 1 Phu Dong Thien Vuong, Dalat, Viet Nam
  • 5 Department of Mathematics, Hanoi Pedagogical University 2, Xuan Hoa, Phuc Yen, Vinh Phuc, Viet Nam
  • 6 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, China
Existence theorems; Geoffrion-properly efficient solutions; M-tameness; Palais–Smale conditions; Pareto efficient solutions; Properness
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