羅勝欣，葉冬平

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魯棒錐規(guī)劃的Fenchel-Lagrange強(qiáng)對(duì)偶

羅勝欣，葉冬平

（吉首大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)院，湖南 吉首 416000 ）

利用共軛函數(shù)的上圖性質(zhì)，引進(jìn)新的魯棒型約束規(guī)劃條件，等價(jià)刻畫了魯棒錐約束優(yōu)化問(wèn)題與其對(duì)偶問(wèn)題之間的Fenchel-Lagrange強(qiáng)對(duì)偶和穩(wěn)定強(qiáng)對(duì)偶，推廣和改進(jìn)了前人的相關(guān)結(jié)論。

Fenchel-lagrange強(qiáng)對(duì)偶； 魯棒型約束條件； 錐規(guī)劃

0．引言

約束優(yōu)化問(wèn)題的研究是現(xiàn)代優(yōu)化理論中的重要課題之一，受到了學(xué)者們的廣泛關(guān)注。特別地，許多學(xué)者研究了經(jīng)典的錐約束優(yōu)化問(wèn)題，建立了錐規(guī)劃的對(duì)偶理論、(KKT)類最優(yōu)性條件、穩(wěn)定性分析等一系列有意義的結(jié)論（參看文獻(xiàn)[1-6]）。例如，文獻(xiàn)[3]利用閉性條件，建立了錐約束優(yōu)化問(wèn)題的強(qiáng)對(duì)偶和穩(wěn)定強(qiáng)對(duì)偶；文獻(xiàn)[1,2]利用上圖類條件和次微分條件，等價(jià)刻畫了錐約束優(yōu)化問(wèn)題的強(qiáng)對(duì)偶、Farkas引理、最優(yōu)性條件和全對(duì)偶等。

與此同時(shí)，上述優(yōu)化模型大都假設(shè)輸入的數(shù)據(jù)是精確的，這種假設(shè)沒(méi)有考慮到模型的質(zhì)量及可行性受到的數(shù)據(jù)不確定性的影響。實(shí)際上，由于測(cè)量誤差或模型本身的缺陷，或者決策階段缺乏信息等原因，許多優(yōu)化問(wèn)題的數(shù)據(jù)是受到干擾的或是不確定的，并且概率分布也無(wú)法預(yù)知。因此，如何在數(shù)據(jù)不確定的情形下給出約束優(yōu)化問(wèn)題的對(duì)偶理論的等價(jià)刻畫，成為了現(xiàn)代優(yōu)化理論研究中的又一個(gè)熱點(diǎn)和難點(diǎn)問(wèn)題。為此，許多學(xué)者研究了如下數(shù)據(jù)不確定下的魯棒錐約束優(yōu)化問(wèn)題

1．預(yù)備知識(shí)

2．魯棒錐規(guī)劃的Fenchel-Lagrange對(duì)偶

及其Fenchel-Lagrange對(duì)偶問(wèn)題

故(3.4)式成立當(dāng)且僅當(dāng)

由此可知結(jié)論成立。

因此式(3.6)成立。

從而由上圖定義有

于是，

由定理3.1及命題3.1可得以下推論。

[1] D. H. Fang, C. Li, K. F. Ng. Constraint qualifications for optimality conditions and total Lagrangian dualities in convex infinite programming [J]. Nonlinear Anal., 2010, 73: 1143-1159.

[2] D. H. Fang, C. Li, K. F. Ng. Constraint qualifications for extended Farkas’s lemmas and Lagrangian dualities in convex infinite programming [J]. SIAM J. Optim., 2009, 20 : 1311-1332.

[3] R. I. Bot, S. M. Grad, G. Wanka. New regularity conditions for strong and total Fenchel-Lagrange duality in infinite dimensional spaces [J]. Nonlinear Anal., 2008, 69: 323-336.

[4] R. I. Bot, S. M. Grad, G. Wanka. A new constraint qualification for the formula of the subdifferential of composed convex functions in infinite dimensional spaces [J]. Math. Nachr., 2008, 281: 1-20.

[5] M. A. Goberna, V. Jeyakumar, M. A. Lopez. Necessary and sufficient conditions for solvability of systems of infinite convex inequalities [J]. Nonlinear Anal., 2008, 68: 1184-1194.

[6] V. Jeyakumar. Characterizing set containments involving infinite convex constraints and reverse-convex constraints [J]. SIAM J. Optim., 2003 ,13: 947-959.

[7] G. Y. Li, V. Jeyakumar and G. M. Lee. Robust conjugate duality for convex optimization under uncertainty with application to data classification [J]. Nonlinear Anal., 2011, 74: 2327-2343.

[8] D. H. Fang, C. Li, J. C. Yao. Stable lagrange dualities for robust conical programming [J]. J. Nonlinear Convex Anal., 2015, 16: 2141-2158.

[9] V. Jeyakumar, G. Y. Li, J. H. Wang. Some robust convex programs without a duality gap [J]. J. convex Anal., 2013, 2: 377-394.

[10] V. Jeyakumar, G. Y. Li, Strong duality in robust convex programming: complete characterizations [J]. SIAM J. Optim., 2010, 20: 3384-3407.

Fenchel-Lagrange Strong Duality for Robust Conical Programming

LUO Shengxin, YE Dongping

( College of Mathematics and Statistics, Jishou University, Jishou 416000, Hunan, China )

In this paper, we introduce some new robust-type constraint qualifications by using the properties of the epigraph of the conjugate functions. Under the new constraint qualifications, the strong duality and the stable strong duality between robust conical optimization problem and its Fenchel-Lagrange dual problem are established, which extend and improve the corresponding results in the previous papers.

Fenchel-Lagrange strong duality, robust-type constraint qualifications, conical programming

O174

A

1673-9639 (2018) 09-0063-04

2018-03-22

國(guó)家自然科學(xué)基金（11461027）；湖南省教育廳科研基金（17A172）。

羅勝欣（1994-），男，湖南邵東人，碩士研究生，研究方向：最優(yōu)化理論與方法。 葉冬平（1994-），男，湖南常德人，碩士研究生，研究方向：最優(yōu)化理論與方法。

（責(zé)任編輯 毛 志）（責(zé)任校對(duì) 印有家）