Comments on: Farkas-lemma: three decades of generalizations for mathematical optimization

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• 作者：B. S. Mordukhovich (1) (2)
  
• 关键词：Variational analysis and optimization ; Farkas-lemma ; Convex programming ; Semi ; infinite programming ; Generalized differentiation ; 49J52 ; 49J53 ; 90C25 ; 90C34
• 刊名：TOP
• 出版年：2014
• 出版时间：April 2014
• 年：2014
• 卷：22
• 期：1
• 页码：31-37
• 全文大小：123 KB
• 参考文献：1. Bartl D (2008) A short algebraic proof of the Farkas lemma. SIAM J Optim 19:234-39 CrossRef
  2. Cánovas MJ, López MA, Mordukhovich BS, Parra J (2009) Variational analysis in semi-infinite and infinite programming, I: stability of linear inequality systems of feasible solutions. SIAM J Optim 20:1504-526 CrossRef
  3. Cánovas MJ, López MA, Mordukhovich BS, Parra J (2010) Variational analysis in semi-infinite and infinite programming, II: necessary optimality conditions. SIAM J Optim 20:2788-806 CrossRef
  4. Cánovas MJ, López MA, Mordukhovich BS, Parra J (2012) Quantitative stability of linear infinite inequality systems under block perturbations with applications to convex systems. TOP 20:310-27 CrossRef
  5. Farkas G (1894) A Fourier-féle mechanikai elv alkamazsai. Mathematikai és Termszettudományi értesto 12:457-72
  6. Henrion R, Mordukhovich BS, Nam NM (2010) Second-order analysis of polyhedral systems in finite and infinite dimensions with applications to robust stability of variational inequalities. SIAM J Optim 20:2199-227 CrossRef
  7. Mordukhovich BS (2006) Variational analysis and generalized differentiation, I: basic theory, II: applications. Springer, Berlin
  8. Mordukhovich BS, Nghia TTA (2013a) Constraint qualifications and optimality conditions in semi-infinite and infinite programming. Math Program 139:271-00 CrossRef
  9. Mordukhovich BS, Nghia TTA (2013b) Subdifferentials of nonconvex supremum functions and their applications to semi-infinite and infinite programs with Lipschitzian data. SIAM J Optim 23:406-31 CrossRef
  10. Mordukhovich BS, Nghia, TTA (2013c) Nonsmooth cone-constrained optimization with applications to semi-infinite programming. Math Oper Res http://dx.doi.org/10.1287/moor.2013.0622
  11. Rockafellar RT, Wets RJB (1998) Variational analysis. Springer, Berlin CrossRef
  
• 作者单位：B. S. Mordukhovich (1) (2)
  
  1. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA
  2. King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia
  
• ISSN：1863-8279

文摘

In these comments on the excellent survey by Dinh and Jeyakumar, we briefly discuss some recently developed topics and results on applications of extended Farkas-lemma(s) and related qualification conditions to problems of variational analysis and optimization, which are not fully reflected in the survey. They mainly concern: Lipschitzian stability of feasible solution maps for parameterized semi-infinite and infinite programs with linear and convex inequality constraints indexed by arbitrary sets; optimality conditions for nonsmooth problems involving such constraints; evaluating various subdifferentials of optimal value functions in DC and bilevel infinite programs with applications to Lipschitz continuity of value functions and optimality conditions; calculating and estimating normal cones to feasible solution sets for nonlinear smooth as well as nonsmooth semi-infinite, infinite, and conic programs with deriving necessary optimality conditions for them; calculating coderivatives of normal cone mappings for convex polyhedra in finite and infinite dimensions with applications to robust stability of parameterized variational inequalities. We also give some historical comments on the original Farkas-papers.