Abstract
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.
| Original language | English |
|---|---|
| Pages (from-to) | 1307-1328 |
| Number of pages | 22 |
| Journal | Journal of Industrial and Management Optimization |
| Volume | 13 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - 2017 Jul 1 |
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All Science Journal Classification (ASJC) codes
- Business and International Management
- Strategy and Management
- Control and Optimization
- Applied Mathematics
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Double well potential function and its optimization in the n-dimensional real space - Part II. / Xla, Yong; Sheu, Ruey Lin; Fang, Shu Cherng; Xing, Wenxun.
In: Journal of Industrial and Management Optimization, Vol. 13, No. 3, 01.07.2017, p. 1307-1328.Research output: Contribution to journal › Article
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
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.
UR - http://www.scopus.com/inward/record.url?scp=85021731822&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85021731822&partnerID=8YFLogxK
U2 - 10.3934/jimo.2016074
DO - 10.3934/jimo.2016074
M3 - Article
AN - SCOPUS:85021731822
VL - 13
SP - 1307
EP - 1328
JO - Journal of Industrial and Management Optimization
JF - Journal of Industrial and Management Optimization
SN - 1547-5816
IS - 3
ER -