Sequential Optimality Conditions for Fractional Optimization with Applications to Vector Optimization

详细信息    查看全文

• 作者：Xiang-Kai Sun (1) (2)
  Xian-Jun Long (1)
  Yi Chai (3)
  
  1. College of Mathematics and Statistics ; Chongqing Technology and Business University ; Chongqing聽 ; 400067 ; China
  2. College of Automation ; Chongqing University ; Chongqing聽 ; 400044 ; China
  3. State Key Laboratory of Power Transmission Equipment and System Security and New Technology ; College of Automation ; Chongqing University ; Chongqing聽 ; 400044 ; China
  
• 关键词：Sequential optimality conditions ; Subdifferential ; Constraint qualification ; Fractional optimization ; Vector optimization ; 90C29 ; 90C32 ; 90C46
• 刊名：Journal of Optimization Theory and Applications
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：164
• 期：2
• 页码：479-499
• 全文大小：235 KB
• 参考文献：1. Giannessi, F., Mastroeni, G., Pellegrini, L.: On the theory of vector optimization and variational inequalities: image space analysis and separation. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria, pp. 141鈥?15. Kluwer, Dordrech (2000) i.org/10.1007/978-1-4613-0299-5" target="_blank" title="It opens in new window">CrossRef
  2. Chen, G.Y., Huang, X.X., Yang, X.Q.: Vector Optimization: Set-Valued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems, vol. 541. Springer, Berlin (2005)
  3. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)
  4. Bo牛, R.I., Grad, A., Wanka, G.: Sequential characterization of solutions in convex composite programming and applications to vector optimization. J. Ind. Manag. Optim. 4, 767鈥?82 (2008) i.org/10.3934/jimo.2008.4.767" target="_blank" title="It opens in new window">CrossRef
  5. Li, S.J., Teo, K.L., Yang, X.Q.: High-order optimality conditions for set-valued optimization. J. Optim. Theory Appl. 137, 533鈥?53 (2008) i.org/10.1007/s10957-007-9345-3" target="_blank" title="It opens in new window">CrossRef
  6. Lee, G.M., Kim, G.S., Dinh, N.: Optimality conditions for approximate solutions of convex semi-infinite vector optimization problems. In: Ansar, Q.H., Yao, J.-C. (eds.) Recent Developments in Vector Optimization, Vector Optimization, vol. 1, pp. 275鈥?95. Springer, Berlin (2012) i.org/10.1007/978-3-642-21114-0_8" target="_blank" title="It opens in new window">CrossRef
  7. Jeyakumar, V., Dinh, N., Lee, G.M.: A new closed cone constraint qualification for convex optimization. Applied Mathematics Report AMR 04/8, University of New South Wales (2004)
  8. Burachik, R.S., Jeyakumar, V.: A new geometric condition for Fenchel鈥檚 duality in infinite dimensional spaces. Math. Program. 104, 229鈥?33 (2005) i.org/10.1007/s10107-005-0614-3" target="_blank" title="It opens in new window">CrossRef
  9. Fang, D.H., Li, C., Ng, K.F.: Constraint qualifications for extended Farkas鈥檚 lemmas and Lagrangian dualities in convex infinite programming. SIAM J. Optim. 20, 1311鈥?332 (2009) i.org/10.1137/080739124" target="_blank" title="It opens in new window">CrossRef
  10. Bo牛, R.I.: Conjugate Duality in Convex Optimization. Springer, Berlin (2010)
  11. Jeyakumar, V., Lee, G.M., Dinh, N.: New sequential Lagrange multiplier conditions characterizing optimality without constraint qualifications for convex programs. SIAM J. Optim. 14, 534鈥?47 (2003) i.org/10.1137/S1052623402417699" target="_blank" title="It opens in new window">CrossRef
  12. Jeyakumar, V., Wu, Z.Y., Lee, G.M., Dinh, N.: Liberating the subgradient optimality conditions from constraint qualifications. J. Glob. Optim. 36, 127鈥?37 (2006) i.org/10.1007/s10898-006-9003-6" target="_blank" title="It opens in new window">CrossRef
  13. Bai, F.S., Wu, Z.Y., Zhu, D.L.: Sequential Lagrange multiplier condition for inline-equation id-i-eq575"> ion-source format-t-e-x">\(\varepsilon \) -optimal solution in convex programming. Optimization 57, 669鈥?80 (2008) i.org/10.1080/02331930802355234" target="_blank" title="It opens in new window">CrossRef
  14. Bo牛, R.I., Csetnek, E.R., Wanka, G.: Sequential optimality conditions in convex programming via perturbation approach. J. Convex Anal. 15, 149鈥?64 (2008)
  15. Bo牛, R.I., Csetnek, E.R., Wanka, G.: Sequential optimality conditions for composed convex optimization problems. J. Math. Anal. Appl. 342, 1015鈥?025 (2008) i.org/10.1016/j.jmaa.2007.12.066" target="_blank" title="It opens in new window">CrossRef
  16. Kim, G.S., Lee, G.M.: On inline-equation id-i-eq577"> ion-source format-t-e-x">\(\varepsilon \) -optimality theorems for convex vector optimization problems. J. Nonlinear Convex Anal. 12, 473鈥?82 (2011)
  17. Dinkelbach, W.: On nonlinear fractional programming. Manag. Sci. 13, 492鈥?98 (1967) i.org/10.1287/mnsc.13.7.492" target="_blank" title="It opens in new window">CrossRef
  18. Schaible, S.: Fractional programming, II. On Dinkelbach鈥檚 algorithm. Manag. Sci. 22, 868鈥?73 (1976) i.org/10.1287/mnsc.22.8.868" target="_blank" title="It opens in new window">CrossRef
  19. Mishra, S.K.: Generalized fractional programming problems containing locally subdifferentiable inline-equation id-i-eq579"> ion-source format-t-e-x">\(\rho \) -univex functions. Optimization 41, 135鈥?58 (1997) i.org/10.1080/02331939708844331" target="_blank" title="It opens in new window">CrossRef
  20. Yang, X.M., Teo, K.L., Yang, X.Q.: Symmetric duality for a class of nonlinear fractional programming problems. J. Math. Anal. Appl. 271, 7鈥?5 (2002) i.org/10.1016/S0022-247X(02)00042-2" target="_blank" title="It opens in new window">CrossRef
  21. Bo牛, R.I., Hodrea, I.B., Wanka, G.: Farkas-type results for fractional programming problems. Nonlinear Anal. 67, 1690鈥?703 (2007) i.org/10.1016/j.na.2006.07.041" target="_blank" title="It opens in new window">CrossRef
  22. Sun, X.K., Chai, Y.: On robust duality for fractional programming with uncertainty data. Positivity 18, 9鈥?8 (2014) i.org/10.1007/s11117-013-0227-7" target="_blank" title="It opens in new window">CrossRef
  23. Brondsted, A., Rockafellar, R.T.: On the subdifferential of convex functions. Proc. Am. Math. Soc. 16, 605鈥?11 (1965) i.org/10.1090/S0002-9939-1965-0178103-8" target="_blank" title="It opens in new window">CrossRef
  24. Thibault, L.: Sequential convex subdifferential calculus and sequential Lagrange multipliers. SIAM J. Optim. 35, 1434鈥?444 (1997) i.org/10.1137/S0363012995287714" target="_blank" title="It opens in new window">CrossRef
  25. Bo牛, R.I., Grad, S.M., Wanka, G.: On strong and total Lagrange duality for convex optimization problems. J. Math. Anal. Appl. 337, 1315鈥?325 (2008) i.org/10.1016/j.jmaa.2007.04.071" target="_blank" title="It opens in new window">CrossRef
  26. Li, C., Fang, D.H., L贸pez, G., L贸pez, M.A.: Stable and total Fenchel duality for convex optimization problems in locally convex spaces. SIAM J. Optim. 20, 1032鈥?051 (2009) i.org/10.1137/080734352" target="_blank" title="It opens in new window">CrossRef
  27. Fang, D.H., Li, C., Yang, X.Q.: Stable and total Fenchel duality for DC optimization problems in locally convex spaces. SIAM J. Optim. 21, 730鈥?60 (2011) i.org/10.1137/100789749" target="_blank" title="It opens in new window">CrossRef
  28. Sun, X.K., Li, S.J., Zhao, D.: Duality and Farkas-type results for DC infinite programming with inequality constraints Taiwan. J. Math. 17, 1227鈥?244 (2013)
  
• 刊物主题：Calculus of Variations and Optimal Control; Optimization; Optimization; Theory of Computation; Applications of Mathematics; Engineering, general; Operations Research/Decision Theory;
• 出版者：Springer US
• ISSN：1573-2878

文摘

In this paper, in the absence of any constraint qualifications, a sequential Lagrange multiplier rule condition characterizing optimality for a fractional optimization problem is obtained in terms of the \(\varepsilon \) -subdifferentials of the functions involved at the minimizer. The significance of this result is that it yields the standard Lagrange multiplier rule condition for the fractional optimization problem under a simple closedness condition that is much weaker than the well-known constraint qualifications. A sequential condition characterizing optimality involving only subdifferentials at nearby points to the minimizer is also investigated. As applications, the proposed approach is applied to investigate sequential optimality conditions for vector fractional optimization problems.