TY - JOUR
T1 - Double well potential function and its optimization in the n-dimensional real space - Part II
AU - Xla, Yong
AU - Sheu, Ruey Lin
AU - Fang, Shu Cherng
AU - Xing, Wenxun
N1 - Funding Information:
This research was undertaken while Y. Xia visited the Na tional Cheng Kung University, Tainan, Taiwan. S-C Fang's research has been supported by US ARO Grant # W911NF-15-1-0223. R-L Sheu's research work was sponsored partially by Taiwan NSC 98-2115-M-006-010-MY2 and by National Center for Theoretic Sciences (The southern branch). Y. Xia's research work was supported by National Natural Science Foundation of China under grants 11471325 and 11571029, and by fundamental research funds for the Central Universities un der grant YWF-16-BJ-Y-11. W. Xing's research work was supported by NSFC No. 11171177.
PY - 2017/7/1
Y1 - 2017/7/1
N2 - In contrast to taking the dual approach for finding a global min imum solution of a double well potential function, in Part II of the paper, we characterize the local minimizer, local maximizer, and global minimizer di rectly from the primal side. It is proven that, for a "nonsingular" double well function, there exists at most one local, but non-global, minimizer and at most one local maximizer. Moreover, the local maximizer is "surrounded" by local minimizers in the sense that the norm of the local maximizer is strictly less than that of any local minimizer. We also establish necessary and sufficient optimality conditions for the global minimizer, local non-global minimizer and local maximizer by studying a convex secular function over specific intervals. These conditions lead to three algorithms for identifying different types of crit ical points of a given double well function.
AB - In contrast to taking the dual approach for finding a global min imum solution of a double well potential function, in Part II of the paper, we characterize the local minimizer, local maximizer, and global minimizer di rectly from the primal side. It is proven that, for a "nonsingular" double well function, there exists at most one local, but non-global, minimizer and at most one local maximizer. Moreover, the local maximizer is "surrounded" by local minimizers in the sense that the norm of the local maximizer is strictly less than that of any local minimizer. We also establish necessary and sufficient optimality conditions for the global minimizer, local non-global minimizer and local maximizer by studying a convex secular function over specific intervals. These conditions lead to three algorithms for identifying different types of crit ical points of a given double well function.
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U2 - 10.3934/jimo.2016074
DO - 10.3934/jimo.2016074
M3 - Article
AN - SCOPUS:85021731822
SN - 1547-5816
VL - 13
SP - 1307
EP - 1328
JO - Journal of Industrial and Management Optimization
JF - Journal of Industrial and Management Optimization
IS - 3
ER -