# Saddle points criteria via a second order $\eta$-approximation approach for nonlinear mathematical programming involving second order invex functions

Kybernetika (2011)

• Volume: 47, Issue: 2, page 222-240
• ISSN: 0023-5954

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## Abstract

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In this paper, by using the second order $\eta$-approximation method introduced by Antczak [3], new saddle point results are obtained for a nonlinear mathematical programming problem involving second order invex functions with respect to the same function $\eta$. Moreover, a second order $\eta$-saddle point and a second order $\eta$-Lagrange function are defined for the so-called second order $\eta$-approximated optimization problem constructed in this method. Then, the equivalence between an optimal solution in the original mathematical programming problem and a second order $\eta$-saddle point of the second order $\eta$

## How to cite

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Antczak, Tadeusz. "Saddle points criteria via a second order $\eta$-approximation approach for nonlinear mathematical programming involving second order invex functions." Kybernetika 47.2 (2011): 222-240. <http://eudml.org/doc/196550>.

@article{Antczak2011,
abstract = {In this paper, by using the second order $\eta$-approximation method introduced by Antczak [3], new saddle point results are obtained for a nonlinear mathematical programming problem involving second order invex functions with respect to the same function $\eta$. Moreover, a second order $\eta$-saddle point and a second order $\eta$-Lagrange function are defined for the so-called second order $\eta$-approximated optimization problem constructed in this method. Then, the equivalence between an optimal solution in the original mathematical programming problem and a second order $\eta$-saddle point of the second order $\eta$},
journal = {Kybernetika},
keywords = {second order $\eta$-approximated optimization problem; second order $\eta$-saddle point; second order $\eta$-Lagrange function; second order invex function with respect to $\eta$; second order optimality conditions; second order optimality conditions; second order -approximated optimization problem; second order -saddle point; second order -Lagrange function; second order invex function with respect to },
language = {eng},
number = {2},
pages = {222-240},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Saddle points criteria via a second order $\eta$-approximation approach for nonlinear mathematical programming involving second order invex functions},
url = {http://eudml.org/doc/196550},
volume = {47},
year = {2011},
}

TY - JOUR
TI - Saddle points criteria via a second order $\eta$-approximation approach for nonlinear mathematical programming involving second order invex functions
JO - Kybernetika
PY - 2011
PB - Institute of Information Theory and Automation AS CR
VL - 47
IS - 2
SP - 222
EP - 240
AB - In this paper, by using the second order $\eta$-approximation method introduced by Antczak [3], new saddle point results are obtained for a nonlinear mathematical programming problem involving second order invex functions with respect to the same function $\eta$. Moreover, a second order $\eta$-saddle point and a second order $\eta$-Lagrange function are defined for the so-called second order $\eta$-approximated optimization problem constructed in this method. Then, the equivalence between an optimal solution in the original mathematical programming problem and a second order $\eta$-saddle point of the second order $\eta$
LA - eng
KW - second order $\eta$-approximated optimization problem; second order $\eta$-saddle point; second order $\eta$-Lagrange function; second order invex function with respect to $\eta$; second order optimality conditions; second order optimality conditions; second order -approximated optimization problem; second order -saddle point; second order -Lagrange function; second order invex function with respect to
UR - http://eudml.org/doc/196550
ER -

## References

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1. Antczak, T., 10.1081/NFA-200042183, Numer. Funct. Anal. Optim. 25 (2004), 5–6, 423–438. (2004) MR2106268DOI10.1081/NFA-200042183
2. Antczak, T., 10.1007/s10957-006-9069-9, J. Optim. Theory Appl. 132 (2007), 1, 71–87. (2007) MR2303801DOI10.1007/s10957-006-9069-9
3. Antczak, T., 10.1080/01630560701190265, Numer. Funct. Anal. Optim. 28 (2007), 1–2, 1–13. (2007) Zbl1141.90538MR2302701DOI10.1080/01630560701190265
4. Antczak, T., 10.1007/s10492-009-0028-2, Appl. Math. 54 (2009), 433–445. (2009) Zbl1212.90307MR2545410DOI10.1007/s10492-009-0028-2
5. Bazaraa, M. S., Sherali, H. D., Shetty, C. M., Nonlinear Programming: Theory and Algorithms, John Wiley and Sons, New York 1991. (1991) MR2218478
6. Bector, C. R., Bector, B. K., (Generalized)-bonvex functions and second order duality for a nonlinear programming problem, Congr. Numer. 52 (1985), 37–52. (1985)
7. Bector, C. R., Bector, B. K., On various duality theorems for second order duality in nonlinear programming, Cahiers Centre Études Rech. Opér. 28 (1986), 283–292. (1986) Zbl0622.90068MR0885768
8. Bector, C. R., Chandra, S., Generalized Bonvex Functions and Second Order Duality in Mathematical Programming, Research Report No. 85-2, Department of Actuarial and Management Sciences, University of Manitoba, Winnepeg, Manitoba 1985. (1985)
9. Ben-Tal, A., 10.1007/BF00934107, J. Optim. Theory Appl. 31 (1980), 2, 143–165. (1980) Zbl0416.90062MR0600379DOI10.1007/BF00934107
10. Craven, B. D., 10.1017/S0004972700004895, Bull. Austral. Math. Soc. 24 (1981), 357–366. (1981) Zbl0452.90066MR0647362DOI10.1017/S0004972700004895
11. Hanson, M. A., 10.1016/0022-247X(81)90123-2, J. Math. Anal. Appl. 80 (1981) 545–550. (1981) Zbl0463.90080MR0614849DOI10.1016/0022-247X(81)90123-2
12. Mangasarian, O. L., Nonlinear Programming, McGraw-Hill, New York 1969. (1969) Zbl0194.20201MR0252038
13. Rockafellar, R. T., Convex Analysis, Princeton University Press, 1970. (1970) Zbl0193.18401MR0274683

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