Strict minimizers of order m in nonsmooth optimization problems

Tadeusz Antczak; Krzysztof Kisiel

Commentationes Mathematicae Universitatis Carolinae (2006)

  • Volume: 47, Issue: 2, page 213-232
  • ISSN: 0010-2628

Abstract

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In the paper, some sufficient optimality conditions for strict minima of order m in constrained nonlinear mathematical programming problems involving (locally Lipschitz) ( F , ρ ) -convex functions of order m are presented. Furthermore, the concept of strict local minimizer of order m is also used to state various duality results in the sense of Mond-Weir and in the sense of Wolfe for such nondifferentiable optimization problems.

How to cite

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Antczak, Tadeusz, and Kisiel, Krzysztof. "Strict minimizers of order $m$ in nonsmooth optimization problems." Commentationes Mathematicae Universitatis Carolinae 47.2 (2006): 213-232. <http://eudml.org/doc/249854>.

@article{Antczak2006,
abstract = {In the paper, some sufficient optimality conditions for strict minima of order $m$ in constrained nonlinear mathematical programming problems involving (locally Lipschitz) $(F,\rho )$-convex functions of order $m$ are presented. Furthermore, the concept of strict local minimizer of order $m$ is also used to state various duality results in the sense of Mond-Weir and in the sense of Wolfe for such nondifferentiable optimization problems.},
author = {Antczak, Tadeusz, Kisiel, Krzysztof},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {nonsmooth programming; strict local minimizer of order $m$; Clarke’s generalized gradient; $(F, \rho )$-convex function of order $m$ with respect to $\theta $; nonsmooth programming; strict local minimizer of order },
language = {eng},
number = {2},
pages = {213-232},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Strict minimizers of order $m$ in nonsmooth optimization problems},
url = {http://eudml.org/doc/249854},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Antczak, Tadeusz
AU - Kisiel, Krzysztof
TI - Strict minimizers of order $m$ in nonsmooth optimization problems
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 2
SP - 213
EP - 232
AB - In the paper, some sufficient optimality conditions for strict minima of order $m$ in constrained nonlinear mathematical programming problems involving (locally Lipschitz) $(F,\rho )$-convex functions of order $m$ are presented. Furthermore, the concept of strict local minimizer of order $m$ is also used to state various duality results in the sense of Mond-Weir and in the sense of Wolfe for such nondifferentiable optimization problems.
LA - eng
KW - nonsmooth programming; strict local minimizer of order $m$; Clarke’s generalized gradient; $(F, \rho )$-convex function of order $m$ with respect to $\theta $; nonsmooth programming; strict local minimizer of order
UR - http://eudml.org/doc/249854
ER -

References

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  1. Auslender A., Stability in mathematical programming with nondifferentiable data, SIAM J. Control Optim. 22 (1984), 239-254. (1984) Zbl0538.49020MR0732426
  2. Bazaraa M.S., Sherali H.D., Shetty C.M., Nonlinear Programming: Theory and Algorithms, John Wiley and Sons, New York, 1991. Zbl1140.90040MR2218478
  3. Ben-Israel A., Mond B., What is invexity?, J. Austral. Math. Soc. Ser. B 28 (1986), 1-9. (1986) Zbl0603.90119MR0846778
  4. Clarke F.H., Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, 1983. Zbl0696.49002MR0709590
  5. Craven B.D., Nonsmooth multiobjective programming, Numer. Funct. Anal. Optim. 10 1-2 (1989), 49-64. (1989) Zbl0645.90076MR0978802
  6. Cromme L., Strong uniqueness: a far-reaching criterion for the convergence of iterative procedures, Numer. Math. 29 (1978), 179-193. (1978) MR0461890
  7. Egudo R.R., Mond B., Duality with generalized convexity, J. Austral. Math. Soc. Ser. B 28 (1986), 10-21. (1986) Zbl0608.49012MR0846779
  8. Hiriart-Urruty J.-B., Refinements of necessary optimality conditions in nondifferentiable programming I, Appl. Math. Optim. 5 (1979), 63-82. (1979) Zbl0389.90088MR0526428
  9. Hanson M.A., On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl. 80 (1981), 545-550. (1981) Zbl0463.90080MR0614849
  10. Hanson M.A., Mond B., Further generalizations of convexity in mathematical programming, J. Inform. Optim. Sci. 3 (1982), 25-32. (1982) Zbl0475.90069MR0713163
  11. Jeyakumar V., Strong and weak invexity in mathematical programming, Math. Oper. Res. 55 (1985), 109-125. (1985) Zbl0566.90086MR0811672
  12. Jeyakumar V., Equivalence of a saddle-points and optima, and duality for a class of non-smooth non-convex problems, J. Math. Anal. Appl. 130 (1988), 334-343. (1988) MR0929939
  13. Kaul R.N., Suneja S.K., Lalitha C.S., Generalized nonsmooth invexity, J. Inform. Optim. Sci. 15 (1994), 1-17. (1994) Zbl0852.90113MR1262012
  14. Klatte D., Stable local minimizers in semi-infinite optimization: regularity and second-order conditions, J. Comput. Appl. Math. 56 (1994), 137-157. (1994) Zbl0823.90121MR1338641
  15. Mangasarian O.L., Nonlinear Programming, McGraw-Hill, New York, 1969. Zbl0833.90108MR0252038
  16. Mond B., Weir T., Generalized concavity and duality, in: Generalized Concavity in Optimization and Economics, edited by S. Schaible and W.T. Ziemba, Academic Press, New York, 1981, pp.263-279. Zbl0619.90062MR0652702
  17. Preda V., On efficiency and duality for multiobjective programs, J. Math. Anal. Appl. 166 (1992), 365-377. (1992) Zbl0764.90074MR1160932
  18. Singer I., Abstract Convex Analysis, John Wiley and Sons, New York, 1997, . MR1461544
  19. Studniarski M., Necessary and sufficient conditions for isolated local minima of nonsmooth functions, SIAM J. Control Optim. 24 (1986), 1044-1049. (1986) Zbl0604.49017MR0854069
  20. Studniarski M., Sufficient conditions for the stability of local minimum points in nonsmooth optimization, Optimization 20 (1989), 27-35. (1989) Zbl0679.90072MR0977217
  21. Studniarski M., Characterizations of strict local minima for some nonlinear programming problems, Nonlinear Anal. 30 (1997), 5363-5367 (Proc. 2nd World Congress of Nonlinear Analysts). (1997) Zbl0914.90243MR1726039
  22. Ward D.E., Characterizations of strict local minima and necessary conditions for weak sharp minima, J. Optim. Theory Appl. 80 (1994), 551-571. (1994) Zbl0797.90101MR1265176
  23. Wolfe P., A duality theorem for nonlinear programming, Quart. Appl. Math. 19 (1961), 239-244. (1961) MR0135625

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