Skip to main content
SpringerLink
Log in

Increasing-along-rays property, vector optimization and well-posedness

  • Original Article
  • Published:
Mathematical Methods of Operations Research Aims and scope Submit manuscript

Abstract

In this paper we consider three classes of increasing-along-rays maps. We investigate the relations between increasing-along-rays property and star- shaped vector optimization. We also study well-posedness issues in star-shaped vector optimizations associated with increasing-along-rays maps.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Bednarczuk E (1987) Well posedness of vector optimization proble. In: Jahn J, Krabs W (eds) Recent advances and historical development of vector optimization problems, Lecture Notes in Economics and Mathematical systems, vol 294. Springer, Berlin Heidelberg New York, pp 51–61

    Google Scholar 

  • Bednarczuk E (1994) An approach to well-posedness in vector optimization: consequences to stability, Parametric optimization. Control Cybernet 23(1–2):107–122

    MATH  MathSciNet  Google Scholar 

  • Bianchi M, Hadjisavvas N, Schaible S (1997) Vector equilibrium problems with generalized monotone bifunctions. J Optim Theory Appl 92(3):527–542

    Article  MATH  MathSciNet  Google Scholar 

  • Chen GY, Yang XQ (1990) The vector complementary problem and its equivalences with the weak minimal elements in ordered spaces. J Math Anal Appl 153(1):136–158

    Article  MATH  MathSciNet  Google Scholar 

  • Crespi GP, Ginchev I, Rocca M (2004) Minty variational inequalities, increase-along-rays property and optimization. J Optim Theory Appl 123(3):479–496

    Article  MATH  MathSciNet  Google Scholar 

  • Crespi GP, Ginchev I, Rocca M (2005) Existence of solutions and star-shapedness in Minty variational inequalities. J Global Optim 32(4):485–494

    Article  MATH  MathSciNet  Google Scholar 

  • Dentcheva D, Helbig S (1996) On variational principles, level sets, well-posedness, and ε-solutions in vector optimization. J Optim Theory Appl 89(2):325–349

    Article  MATH  MathSciNet  Google Scholar 

  • Dontchev AL, Zolezzi T (1993) Well-posed optimization problems, Lecture Notes in Math. 1543 Springer, Berlin Heidelberg New York

  • Guo DJ, Cho YJ, Zhu J (2004) Partial Ordering Methods in Nonlinear Problems. Nova Science Publishers, New York

    MATH  Google Scholar 

  • Huang XX (2000a) Extended well-posedness properties of vector optimization problems. J Optim Theory Appl 106(1):165–182

    Article  MATH  MathSciNet  Google Scholar 

  • Huang XX (2000b) Stability in vector-valued and set-valued optimization. Math Methods Oper Res 52(2):185–193

    Article  MATH  MathSciNet  Google Scholar 

  • Huang XX (2001) Pointwise well-posedness of perturbed vector optimization problems in a vector-valued variational principle. J Optim Theory Appl 108(3):671–684

    Article  MATH  MathSciNet  Google Scholar 

  • Lemaire B, Ould Ahmed Salem C, Revalski JP (2002) Well-posedness by perturbations of variational problems. J Optim Theory Appl 115(2):345–368

    Article  MATH  MathSciNet  Google Scholar 

  • Loridan P (1995) Well-posedness in vector optimization. In: Lucchetti R, Revalski J (eds) Recent developments in variational well-posedness problems. Mathematics and its Applications, vol 331. Kluwer Academic Publisher, Dordrecht, pp 171–192

    Google Scholar 

  • Luc DT (1989) Theory of Vector Optimization, Leture Notes in Economics and Mathematical Systems, No. 319, Springer, Berlin Heidelberg New York

  • Lucchetti R (1987) Well posedness, towards vector optimization. In: Jahn J, Krabs W (eds) Recent advances and historical development of vector optimization problems, Lecture Notes in Economics and Mathematical systems, vol 294. Springer, Berlin Heidelberg New York, pp 194–207

    Google Scholar 

  • Miglierina E, Molho E (2003) Well-posedness and convexity in vector optimization. Math Methods Oper Res 58(3):375–385

    Article  MATH  MathSciNet  Google Scholar 

  • Miglierina E, Molho E, Rocca M (2005) Well-posedness and scalarization in vector optimization. J Optimi Theory Appl 126(2):391–409

    Article  MATH  MathSciNet  Google Scholar 

  • Penot JP (1997) Duality for radiant and shady programs. Acta Math Vietnam 22(2):541–566

    MATH  MathSciNet  Google Scholar 

  • Rubinov AM (2000) Abstract Convexity and Global Optimization. Kluwer Acad Publ, Dordrecht

    MATH  Google Scholar 

  • Rubinov AM, Andramonov MY (1999) Minimizing increasing star-shaped functions based on abstract convexity. J Global Optim 15(1):19–39

    Article  MATH  MathSciNet  Google Scholar 

  • Shveidel A (1997) Separability of star-shaped sets and its application to an optimization problem. Optimization 40(3):207–227

    MATH  MathSciNet  Google Scholar 

  • Tanaka T (1997) Generalized semicontinuity and existence theorems for cone saddle points. Appl Math Optim 36(3):313–322

    Article  MATH  MathSciNet  Google Scholar 

  • Tykhonov AN (1966) On the stability of the functional optimization problem. USSR J Comput Math Math Physics 6:631–634

    Google Scholar 

  • Zaffaroni A (2004) Is every radiant function the sum of quasiconvex functions? Math Methods Oper Res 59(2):221–233

    Article  MATH  MathSciNet  Google Scholar 

  • Zolezzi T (1995) Well-posedness criteria in optimization with application to the calculus of variations. Nonlinear Anal 25(5):437–453

    Article  MATH  MathSciNet  Google Scholar 

  • Zolezzi T (1996) Extended well-posedness of optimization problems. J Optim Theory Appl 91(1):257–266

    Article  MATH  MathSciNet  Google Scholar 

  • Zolezzi T (2001) Well-posedness and optimization under perturbations. Optimization with data perturbations. Ann Oper Res 101:351–361

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, People’s Republic of China

    Ya-ping Fang & Nan-jing Huang

Authors
  1. Ya-ping Fang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nan-jing Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ya-ping Fang.

Additional information

This work is supported by the Basic and Applied Research Projection of Sichuan Province (05JY029-009-1) and the National Natural Science Foundation of China.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fang, Yp., Huang, Nj. Increasing-along-rays property, vector optimization and well-posedness. Math Meth Oper Res 65, 99–114 (2007). https://doi.org/10.1007/s00186-006-0113-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00186-006-0113-1

Keywords

Mathematics Subject Classification (2000)

Search

Navigation