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Mathematical Methods of Operations Research

, Volume 43, Issue 3, pp 319–336 | Cite as

Strengthened invex and perturbations

  • Bruce Desmond Craven
  • Do Van Luu
  • Bevil Milton Glover
Article

Abstract

If the strengthened invex property holds for a constrained minimization problem, then a Karush-Kuhn-Tucker point is a strict minimum. The strict minimum property is preserved under small perturbations of the problem. This allows sufficient conditions to be given for a minimax, starting from Karush-Kuhn-Tucker conditions. They extend to vector-valued minimax and to nonsmooth (Lipschitz) problems. An example is provided to illustrate the strengthened invex property, also a discussion of quadratic-linear (nonconvex) programming implications.

Key words

Strict constrained minimum strengthened invexity stability to perturbations minimax nonsmooth analysis 

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References

  1. [1]
    Attouch H, Wets RJB (1993) Quantitative stability of variational systems II. A framework for nonlinear conditioning. SIAM J Optimization 3/2:359–381Google Scholar
  2. [2]
    Bolintineanu S, Craven BD (1992) Linear multicriteria sensitivity and shadow costs. Optimization 26:115–127Google Scholar
  3. [3]
    Chandra S (1972) Strong pseudoconvex programming. Indian J of Pure and Applied Math 3:178–182Google Scholar
  4. [4]
    Craven BD (1980) Strong vector minimization and duality. ZAMM 60:1–5Google Scholar
  5. [5]
    Craven BD (1981) Invex functions and constrained local minima. Bulletin of the Australian Mathematical Society 24:357–366Google Scholar
  6. [6]
    Craven BD (1988) Lagrangian conditions for a minimax. Proceedings of Conference on Functional Analysis/Optimization. Australian National University, Canberra 24–33Google Scholar
  7. [7]
    Craven BD (1989) A modified wolfe dual for weak vector minimization. Numerical Functional Analysis and Optimization 10:899–907Google Scholar
  8. [8]
    Craven BD (1994) Convergence of discrete approximations for constrained minimization. J Austral Math Soc B 35:1–12Google Scholar
  9. [9]
    Craven BD, Glover BM (1985) Invex functions and duality. J Austral Math Soc A 39:1–20Google Scholar
  10. [10]
    Craven BD, Luu DV (1993) An approach to optimality conditions for nonsmooth minimax problems. University of Melbourne, Mathematics Department, Preprint Series 8Google Scholar
  11. [11]
    Craven BD, Luu DV (1993) Constrained minimax for a vector-valued function. University of Melbourne, Mathematics Department, Preprint Series 11Google Scholar
  12. [12]
    Jeyakumar V (1985) Strong and weak invexity in mathematical programming. Methods of Operations Research 55:109–125Google Scholar
  13. [13]
    Jeyakumar V (1986) On subgradient duality with strong and weak convex functions. J Austral Math Soc A 40:143–152Google Scholar
  14. [14]
    Jeyakumar V (1986) ϱ-convexity and second order duality. Utilitas Mathematica 29:71–85Google Scholar
  15. [15]
    Jeyakumar V (1988) Equivalence of saddlepoints and optima, and duality for a class of non-smooth nonconvex problems. J Math Analysis Appl 130:334–343Google Scholar
  16. [16]
    Mordukhovich B (1992) Stability theory for parametric generalized equations and variational inequalities in nonsmooth analysis. IMA, Preprint Series 997Google Scholar
  17. [17]
    Vial JP (1982) Strong convexity of sets and functions. J Math Economics 9:187–205Google Scholar
  18. [18]
    Vial JP (1983) Strong and weak convexity of sets and functions. Math Oper Res 8:231–239Google Scholar

Copyright information

© Physica-Verlag 1996

Authors and Affiliations

  • Bruce Desmond Craven
    • 1
  • Do Van Luu
    • 2
  • Bevil Milton Glover
    • 3
  1. 1.Mathematics DepartmentUniversity of MelbourneParkvilleAustralia
  2. 2.Institute of Mathematics, HanoiBo Ho, HanoiVietnam
  3. 3.School of Information Technology and Mathematical SciencesUniversity of BallaratBallaratAustralia

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