First order solutions in conic programming

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• 作者：Mirjam Dür ; Bolor Jargalsaikhan ; Georg Still
• 关键词：Conic programs ; First order maximizer ; Linear semi ; infinite programming ; Stability of first order maximizers ; 90C25 ; 90C34
• 刊名：Mathematical Methods of Operations Research (ZOR)
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：82
• 期：2
• 页码：123-142
• 全文大小：543 KB
• 参考文献：Ahmed F, Dür M, Still G (2013) Copositive programming via semi-infinite optimization. J Optim Theory Appl 159:322-40MATH MathSciNet CrossRef
  Alizadeh F, Haeberly JA, Overton ML (1997) Complementarity and nondegeneracy in semidefinite programming. Math Program 77:111-28MATH MathSciNet
  Bonnans JF, Shapiro A (2000) Perturbation analysis of optimization problems. Springer, BerlinMATH CrossRef
  Bolte J, Daniilidis A, Lewis AS (2011) Generic optimality conditions for semialgebraic convex programs. Math Oper Res 36:55-0MATH MathSciNet CrossRef
  Dür M, Jargalsaikhan B, Still G (2014) Genericity results in linear conic programming—a tour d’horizon. Preprint (2014) submitted
  Fischer T (1991) Strong uniqueness and alternation for linear optimization. J Optim Theory Appl 69:251-67MATH MathSciNet CrossRef
  Goberna MA, Lopez MA, Todorov M (1995) Unicity in linear optimization. J Optim Theory Appl 86:37-6MATH MathSciNet CrossRef
  Goberna MA, Lopez MA (1998) Linear semi-infinite optimization. Wiley, New YorkMATH
  Goberna MA, Todorov MI, Vera de Serio VN (2012) On stable uniqueness in linear semi-infinite optimization. J Global Optim 53:347-61MATH MathSciNet CrossRef
  Helbig S, Todorov MI (1998) Unicity results for general linear semi-infinite optimization problems using a new concept of active constraints. Appl Math Optim 38:21-3MATH MathSciNet CrossRef
  Hettich R, Kortanek K (1993) Semi-infinite programming: theory, methods and applications. SIAM Rev 35:380-29MATH MathSciNet CrossRef
  Maxfield JE, Minc H (1962/1963) On the matrix equation \(X^{\prime }X = A\) . Proc Edinburgh Math Soc 13:125-29
  Nürnberger G (1985) Unicity in semi-infinite optimization. In: Brosowski et al. (eds) Parametric optimization and approximation (Oberwolfach 1983). Birkh?user Verlag, Basel, pp 231-47
  Papadopoulou S (1977) On the geometry of stable compact convex sets. Math Ann 229:193-00MATH MathSciNet CrossRef
  Pataki G (2000) The geometry of semidefinite programming. In: Wolkowicz H, Saigal R, Vandenberghe L (eds) Handbook of semidefinite programming. Academic, Dordrecht, pp 29-5CrossRef
  Pataki G (2013) On the connection of facially exposed and nice cones. J Math Anal Appl 400:211-21MATH MathSciNet CrossRef
  Pataki G, Tun?el L (2001) On the generic properties of convex optimization problems in conic form. Math Program 89:449-57MATH MathSciNet CrossRef
  Roshchina V (2014) Facially exposed cones are not always nice. SIAM J Optim 24:257-68MATH MathSciNet CrossRef
  Shapiro A (1997) First and second order analysis of nonlinear semidefinite programs. Math Program 77:301-20
  
• 作者单位：Mirjam Dür (1)
  Bolor Jargalsaikhan (2)
  Georg Still (3)
  
  1. Department of Mathematics, University of Trier, 54286, Trier, Germany
  2. Department of Operations, Faculty of Economics and Business, University of Groningen, P.O. Box 800, 9700 AV, Groningen, The Netherlands
  3. Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE, Enschede, The Netherlands
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Calculus of Variations and Optimal Control
  Operation Research and Decision Theory
  Business
  
• 出版者：Physica Verlag, An Imprint of Springer-Verlag GmbH
• ISSN：1432-5217

文摘

We study the order of maximizers in linear conic programming (CP) as well as stability issues related to this. We do this by taking a semi-infinite view on conic programs: a linear conic problem can be formulated as a special instance of a linear semi-infinite program (SIP), for which characterizations of the stability of first order maximizers are well-known. However, conic problems are highly special SIPs, and therefore these general SIP-results are not valid for CP. We discuss the differences between CP and general SIP concerning the structure and results for stability of first order maximizers, and we present necessary and sufficient conditions for the stability of first order maximizers in CP. Keywords Conic programs First order maximizer Linear semi-infinite programming Stability of first order maximizers