模糊值函數(shù)的凸性與次可微性

關(guān)世霞,包玉娥,趙慧冬

(內(nèi)蒙古民族大學(xué)數(shù)學(xué)學(xué)院,內(nèi)蒙古 通遼 028043)

模糊值函數(shù)的凸性與次可微性

關(guān)世霞,包玉娥,趙慧冬

(內(nèi)蒙古民族大學(xué)數(shù)學(xué)學(xué)院,內(nèi)蒙古 通遼 028043)

在Goetschel-Voxman所定義的序關(guān)系下,首先討論了模糊值函數(shù)的凸性,得到了凸模糊值函數(shù)的若干充分條件,并證明了凸模糊值函數(shù)的Jensen不等式;其次,討論了凸模糊值函數(shù)的次可微性,給出了次微分的若干重要性質(zhì),并得到了次可微條件下取得最優(yōu)解的充分必要條件以及若干個(gè)次可微的充分條件.

模糊值函數(shù);梯度;次微分;最優(yōu)解

1 引言

自從模糊數(shù)學(xué)理論建立以來(lái),由于它在處理廣泛存在的模糊性方面的成功,在某種程度上彌補(bǔ)了經(jīng)典數(shù)學(xué)和統(tǒng)計(jì)數(shù)學(xué)的不足.對(duì)于數(shù)學(xué)規(guī)劃方面的許多實(shí)際問(wèn)題人們也越來(lái)越多地使用了模糊數(shù)學(xué)的知識(shí),并在眾多學(xué)者的共同努力下,模糊規(guī)劃得到了迅速的發(fā)展.在經(jīng)典數(shù)學(xué)中凸性理論在數(shù)學(xué)規(guī)劃中有著非常重要的作用.隨著模糊規(guī)劃研究的深入與發(fā)展,自然想把經(jīng)典數(shù)學(xué)規(guī)劃中的某些方法推廣應(yīng)用到模糊規(guī)劃中,從而也開(kāi)始探討了凸模糊集合和凸模糊值函數(shù)在模糊規(guī)劃中的應(yīng)用,豐富了數(shù)學(xué)規(guī)劃的研究?jī)?nèi)容.

2003年文獻(xiàn)[1]提出了模糊值函數(shù)的方向?qū)?shù)和次微分,梯度與次梯度等概念并應(yīng)用于模糊規(guī)劃中.同樣為了探討模糊規(guī)劃的對(duì)偶理論,2005年文獻(xiàn) [2]也提出了凸模糊值函數(shù)的次梯度,次微分等概念,討論了凸模糊值函數(shù)的極值問(wèn)題.上述工作中由于模糊數(shù)的序結(jié)構(gòu)條件比較強(qiáng),導(dǎo)致經(jīng)典凸分析中的很多結(jié)果在模糊數(shù)空間中無(wú)法推廣和表示.2001年文獻(xiàn)[3]利用Goetschel-Voxman在文獻(xiàn)[4]中所給出的序關(guān)系,討論了模糊值函數(shù)的凸性問(wèn)題,得到了很好的結(jié)果.2010年文獻(xiàn)[5]中,同樣利用此序關(guān)系討論了模糊值函數(shù)的可微性,并利用梯度討論了模糊規(guī)劃及凸模糊規(guī)劃等問(wèn)題,給出了取得最優(yōu)解的一些條件.在文獻(xiàn)[6]中討論了一類廣義凸模糊值函數(shù)的若干性質(zhì).在此基礎(chǔ)上,利用Goetschel-Voxman在文獻(xiàn)[4]中所給出的序關(guān)系,進(jìn)一步討論了模糊值函數(shù)的凸性以及凸模糊值函數(shù)的次可微性等問(wèn)題.

2 預(yù)備知識(shí)

設(shè)u是實(shí)數(shù)集R上的一個(gè)模糊集,如果u滿足下列條件:

(1)u是上半連續(xù)的;

3 模糊數(shù)值函數(shù)的凸性

4 模糊數(shù)值函數(shù)的次可微性

5 結(jié)論

凸模糊值函數(shù)是模糊規(guī)劃中的重要內(nèi)容,從而對(duì)模糊值函數(shù)的凸性和次可微性問(wèn)題的研究不但有深刻的理論意義,而且還有著潛在的應(yīng)用價(jià)值.本文利用Goetschel-Voxman給出的序關(guān)系,首先討論了模糊值函數(shù)的凸性,得到了凸模糊值函數(shù)的一些新性質(zhì),并證明了凸模糊值函數(shù)的Jensen不等式;其次,討論了凸模糊值函數(shù)的次可微性,證明了次微分的若干重要性質(zhì),并將其應(yīng)用于模糊規(guī)劃中,得到了凸模糊值函數(shù)取得最優(yōu)解的充要條件.這些結(jié)論對(duì)凸模糊值函數(shù)的微分理論及其應(yīng)用研究提供了一個(gè)新的思路.

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Convexity and sub-di ff erentiability of fuzzy-valued function

Guan Shixia,Bao Yu′e,Zhao Huidong
(College of Mathematics,Inner Mogolia National University for Nationalities,Tongliao 028043,China)

Under the order relations de fi ned by Goetschel-Voxman.Firstly,we discussed the convexity of fuzzy-valued function,got a number of sufficient conditions about convex fuzzy-valued function,and proved its Jensen inequality.Then,we discussed the sub-di ff erentiability of fuzzy-valued function,and presented many important properties of sub-di ff erential.At the same time,we gained the necessary and sufficient conditions which we obtained the optimum solution under the sub-di ff erential condition and many sufficient conditions of sub-di ff erential.

fuzzy-valued functions,gradient,sub-di ff erential,optimum solution

O159.2

A

1008-5513(2012)05-0676-11

2011-11-16.

內(nèi)蒙古教自然科學(xué)基金(2010MS0119).

關(guān)世霞(1985-),碩士生,研究方向:模糊分析,凸性理論及其應(yīng)用.

2010 MSC:03E72