By Chew Soo Hong, Zheng Quan
ISBN-10: 3540187723
ISBN-13: 9783540187721
ISBN-10: 3642466230
ISBN-13: 9783642466236
This booklet treats the topic of worldwide optimization with minimum regulations at the habit at the aim capabilities. particularly, optimum stipulations have been built for a category of noncontinuous capabilities characterised through their having point units which are powerful. The integration-based process contrasts with current ways which require a point of convexity or differentiability of the target functionality. a few computational effects on a private computing device are presented.
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Additional resources for Integral Global Optimization: Theory, Implementation and Applications
Example text
We have c l +
0. 1. 3. 7 remain valid for c ~ c after applying a simi- lar limit-based argument. 21) are equivalent to the follOWing alternative definitions. 4: Let c ~ c. c) = lim d~c ~ d>c d Hd are called the variance and the modified variance of f over its level set Hc' respectively. 3 Variance Conditions In this subsection. the variance condition and the modified variance conditions will be stated and proved. 9 (Variance Conditions): For Problem (P). the following are equivalent: a. a point x is a global minimum point with c f{x) as the corresponding glo- bal minimum value of f; b.
1 f(x)~1 1 I IIEk[f(x}+~p(x}]~ - IHcns [f(x}+~P(X}]d~l: 1~(Ek) - ~(Hcns)1 I Ek[f(X)+~p(x)]~1 ~ L and Hcns - ~(Hc ~s)1 IIEk[f(x)+~p(x)]~1 ~ I~(i) --k + c 1~(HcnS}-~(Ek} I ~(HcnS) ~ 0 as k ~ 00. 6: Suppose c is the global minimum value of f over S. c;S) = c. 19) Proof: Let {~} be a decreasing sequence which tends to c as k -+ > O. ~ t CD. CD. c;p) ~ li~~nf ~(~)J~[f(X)+~P(X)]~ ~ li~~UP ~(~)J~[f(X)+~(X)]~ ~ • c. Hence. we have the following theorem. 7: The following are equivalent: a. A point x € S is a global minimum with c = f(x) as the corresponding global minimum value of f over S; b.
Integral Global Optimization: Theory, Implementation and Applications by Chew Soo Hong, Zheng Quan
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