# On robustness of set-valued maps and marginal value functions

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2005)

- Volume: 25, Issue: 1, page 59-108
- ISSN: 1509-9407

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topArmin Hoffmann, and Abebe Geletu. "On robustness of set-valued maps and marginal value functions." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 25.1 (2005): 59-108. <http://eudml.org/doc/271544>.

@article{ArminHoffmann2005,

abstract = {The ideas of robust sets, robust functions and robustness of general set-valued maps were introduced by Chew and Zheng [7,26], and further developed by Shi, Zheng, Zhuang [18,19,20], Phú, Hoffmann and Hichert [8,9,10,17] to weaken up the semi-continuity requirements of certain global optimization algorithms. The robust analysis, along with the measure theory, has well served as the basis for the integral global optimization method (IGOM) (Chew and Zheng [7]). Hence, we have attempted to extend the robust analysis of Zheng et al. to that of robustness of set-valued maps with given structures and marginal value functions. We are also strongly convinced that the results of our investigation could open a way to apply the IGOM for the numerical treatment of some class of parametric optimization problems, when global optima are required.},

author = {Armin Hoffmann, Abebe Geletu},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {robust set; robust function; robust set-valued map; marginal value function; piecewise lower (upper) semi-continuous; approximatable function; approximatable set-valued map; regularity condition; extended Mangasarian-Fromovitz constraint qualification},

language = {eng},

number = {1},

pages = {59-108},

title = {On robustness of set-valued maps and marginal value functions},

url = {http://eudml.org/doc/271544},

volume = {25},

year = {2005},

}

TY - JOUR

AU - Armin Hoffmann

AU - Abebe Geletu

TI - On robustness of set-valued maps and marginal value functions

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2005

VL - 25

IS - 1

SP - 59

EP - 108

AB - The ideas of robust sets, robust functions and robustness of general set-valued maps were introduced by Chew and Zheng [7,26], and further developed by Shi, Zheng, Zhuang [18,19,20], Phú, Hoffmann and Hichert [8,9,10,17] to weaken up the semi-continuity requirements of certain global optimization algorithms. The robust analysis, along with the measure theory, has well served as the basis for the integral global optimization method (IGOM) (Chew and Zheng [7]). Hence, we have attempted to extend the robust analysis of Zheng et al. to that of robustness of set-valued maps with given structures and marginal value functions. We are also strongly convinced that the results of our investigation could open a way to apply the IGOM for the numerical treatment of some class of parametric optimization problems, when global optima are required.

LA - eng

KW - robust set; robust function; robust set-valued map; marginal value function; piecewise lower (upper) semi-continuous; approximatable function; approximatable set-valued map; regularity condition; extended Mangasarian-Fromovitz constraint qualification

UR - http://eudml.org/doc/271544

ER -

## References

top- [1] A. Geletu, A Coarse Solution of Generalized Semi-infinite Optimization via Robust Analysis of Marginal Functions and Global Optimization, Phd. Dissertation, Techncal University of Ilmanu, Institute of Mathematics, Department of Operations Research and Stochastics, December 17, 2004.
- [2] A. Geletu and A. Hoffmann, A conceptual method for solving generalized semi-infinite programming problems via global optimization by exact discontinuous penalization, European J. of OR, V. 157 (2004), 3-15. Zbl1106.90068
- [3] J.-P. Aubin and A. Cellina, Differential Inclusions, Springer Verlag, Berlin 1984.
- [4] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Basel 1990.
- [5] B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer, Non-Linear Parametric Optimization, Akademie-Verlag, Berlin 1982. Zbl0502.49002
- [6] M.S. Bazaraa, H.D. Sherali and C.M. Shetty, Nonlinear Programming, 2nd. ed., John Wiley & Sons, Inc. 1993. Zbl0774.90075
- [7] S. Chew and Q. Zheng, Integral Global Optimization, Springer-Verlag, Berlin 1988.
- [8] J. Hichert, Methoden zur Bestimmung des wesentlichen Supremums mit Anwendung in der globalen Optimierung, Phd. Dissertation, TU-Ilmenau, 1999, Berichte aus der Mathematik, Shaker-Verlag, Aachen 2001.
- [9] J. Hichert, A. Hoffmann and H.X. Phú, Convergence speed of an integral method for computing essential supremum, in Developments in Global Optimization, I.M. Bomze, T. Csendes, R. Horst, P.M. Pardalos, Kluwer Academic Publishers, Dodrecht, Boston, London 1997, 153-170.
- [10] J. Hichert, A. Hoffmann, H.X. Phú and R. Reinhardt, A primal-dual integral method in global optimzation, Discuss. Math. Differential Inclusions, Control and Optimization 20 (2) (2000), 257-278. Zbl0977.90050
- [11] W.W. Hogan, Point-to-set maps in mathematical programming, SIAM Review 15 (3) (1973) 591-603. Zbl0256.90042
- [12] S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis, Volume I, Kluwer Academic Publishers 1997. Zbl0887.47001
- [13] H.T. Jongen, J.-J. Rückmann and O. Stein, Generalized semi-infinite optimization: a first order optimality condition and examples, Math. Prog. 83 (1998), 145-158. Zbl0949.90090
- [14] L.W. Kantorowitsch and G.P. Akilow, Funktionalanalysis in normierten Räumen, Akademie-Verlag, Berlin 1978. Zbl0359.46017
- [15] D. Klatte and R. Henrion, Regularity and stability in non-linear semi-infinite optimization, in Semi-infinite Programming, R. Reemtsen and J.-J. Rückmann (eds.), pp. 69-102, Kluwer Academic Pres, 1998. Zbl0911.90330
- [16] M.M. Kostreva and Q. Zheng, Integral global optimization method for solution of nonlinear complementarity problems, J. Global Opt. 5 (1994), 181-193.
- [17] H.X. Phú and A. Hoffmann, Essential supremum and supremum of summable functions, Numerical Functional Analysis and Optimization 17 (1 & 2) (1996), 167-180. Zbl0877.49013
- [18] S. Shi, Q. Zheng and D. Zhuang, On existence of robust minimizers, in The State of the Art in Global Optimization, C.A. Fouldas and P.M. Pardalos (eds.), pp. 47-56, Kluwer Academic Publishers, 1996. Zbl0879.49005
- [19] S. Shi, Q. Zheng and D. Zhuang, Discontinuous robust mappings are approximatable, American Math. Soc. Trans. 347 (12) (1995), 4943-4957. Zbl0860.90129
- [20] S. Shi, Q. Zheng and D. Zhuang, Set valued robust mappings and approximatable mappings, J. Math. Anal. Appl. 183 (1994), 706-726. Zbl0826.90109
- [21] O. Stein, On level sets of marginal functions, Optimization, 48 (2000) 43-67. Zbl0962.90048
- [22] O. Stein, Bi-level Strategies in Semi-infinite Programming, Kluwer Academic Publishers 2003. Zbl1103.90094
- [23] J.-B.H. Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms I, Springer Verlag 1993.
- [24] G.-W. Weber, Generalized Semi-infinite Optimization and Related Topics, Postdoctoral Thesis, Dept. of Mathematics, Darmstadt University of Technology 1999.
- [25] K. Yosida, Functional Analysis, 6th edition, Springer-Verlag, Berlin-Heidelberg-New York 1980.
- [26] Q. Zheng, Integral Global Optimization of Robust Discontinuous Functions, Ph. D. Dissertation, Clemson University, December 1992.
- [27] Q. Zheng and L. Zhang, Global minimization of constrained problems with discontinuous penality functions, Compt. Math. Appl. 37 (1999), 41-58. Zbl0931.90055
- [28] Q. Zheng and D. Zhuang, Integral global minimization: algorithms, implementations and numerical tests, J. Global Optim. 7 (1995), 421-454. Zbl0846.90105
- [29] Q. Zheng and D. Zhuang, The approximation of fixed points of robust mappings, in Advances in Optimization and Approximation, D.-Z. Du & J. Sun (eds.), pp. 376-389, Kluwer Acadmic publishers 1994. Zbl0828.90138