TY - JOUR
T1 - Fully Piecewise Linear Vector Optimization Problems
AU - Zheng, Xi Yin
AU - Yang, Xiaoqi
N1 - Funding Information:
The authors wish to thank the referees for valuable suggestions, which have helped to improve the presentation of this paper. This research was supported by the National Natural Science Foundation of P. R. China (Grant No. 11771384) Project for Innovation Team of Yunnan Province (202005AE160006) and the Research Grants Council of Hong Kong (PolyU 152128/17E).
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2021/8
Y1 - 2021/8
N2 - We distinguish two kinds of piecewise linear functions and provide an interesting representation for a piecewise linear function between two normed spaces. Based on such a representation, we study a fully piecewise linear vector optimization problem with the objective and constraint functions being piecewise linear. To solve this problem, we divide it into some linear subproblems and structure a dimensional reduction method. Under some mild assumptions, we prove that its Pareto (resp., weak Pareto) solution set is the union of finitely many generalized polyhedra (resp., polyhedra), each of which is contained in a Pareto (resp., weak Pareto) face of some linear subproblem. Our main results are even new in the linear case and further generalize Arrow, Barankin and Blackwell’s classical results on linear vector optimization problems in the framework of finite-dimensional spaces.
AB - We distinguish two kinds of piecewise linear functions and provide an interesting representation for a piecewise linear function between two normed spaces. Based on such a representation, we study a fully piecewise linear vector optimization problem with the objective and constraint functions being piecewise linear. To solve this problem, we divide it into some linear subproblems and structure a dimensional reduction method. Under some mild assumptions, we prove that its Pareto (resp., weak Pareto) solution set is the union of finitely many generalized polyhedra (resp., polyhedra), each of which is contained in a Pareto (resp., weak Pareto) face of some linear subproblem. Our main results are even new in the linear case and further generalize Arrow, Barankin and Blackwell’s classical results on linear vector optimization problems in the framework of finite-dimensional spaces.
KW - Pareto solution
KW - Piecewise linear function
KW - Polyhedron
KW - Weak Pareto solution
UR - http://www.scopus.com/inward/record.url?scp=85109318331&partnerID=8YFLogxK
U2 - 10.1007/s10957-021-01889-w
DO - 10.1007/s10957-021-01889-w
M3 - Journal article
AN - SCOPUS:85109318331
SN - 0022-3239
VL - 190
SP - 461
EP - 490
JO - Journal of Optimization Theory and Applications
JF - Journal of Optimization Theory and Applications
IS - 2
ER -