| Title:
|
Saddle points criteria via a second order $\eta $-approximation approach for nonlinear mathematical programming involving second order invex functions (English) |
| Author:
|
Antczak, Tadeusz |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
47 |
| Issue:
|
2 |
| Year:
|
2011 |
| Pages:
|
222-240 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $ (English) |
| Keyword:
|
second order $\eta $-approximated optimization problem |
| Keyword:
|
second order $\eta $-saddle point |
| Keyword:
|
second order $\eta $-Lagrange function |
| Keyword:
|
second order invex function with respect to $\eta $ |
| Keyword:
|
second order optimality conditions |
| MSC:
|
90C26 |
| MSC:
|
90C46 |
| idZBL:
|
Zbl 1242.90171 |
| idMR:
|
MR2828574 |
| . |
| Date available:
|
2011-06-06T14:55:03Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141569 |
| . |
| Reference:
|
[1] Antczak, T.: An $\eta $-approximation approach to nonlinear mathematical programming involving invex functions.Numer. Funct. Anal. Optim. 25 (2004), 5–6, 423–438. MR 2106268, 10.1081/NFA-200042183 |
| Reference:
|
[2] Antczak, T.: Saddle points criteria in an $\eta $-approximation approach for nonlinear mathematical programming involving invex functions.J. Optim. Theory Appl. 132 (2007), 1, 71–87. MR 2303801, 10.1007/s10957-006-9069-9 |
| Reference:
|
[3] Antczak, T.: A modified objective function method in mathematical programming with second order invexity.Numer. Funct. Anal. Optim. 28 (2007), 1–2, 1–13. Zbl 1141.90538, MR 2302701, 10.1080/01630560701190265 |
| Reference:
|
[4] Antczak, T.: A second order $\eta $-approximation method for constrained optimization problems involving second order invex functions.Appl. Math. 54 (2009), 433–445. Zbl 1212.90307, MR 2545410, 10.1007/s10492-009-0028-2 |
| Reference:
|
[5] Bazaraa, M. S., Sherali, H. D., Shetty, C. M.: Nonlinear Programming: Theory and Algorithms.John Wiley and Sons, New York 1991. MR 2218478 |
| Reference:
|
[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. |
| Reference:
|
[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. Zbl 0622.90068, MR 0885768 |
| Reference:
|
[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. |
| Reference:
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[9] Ben-Tal, A.: Second-order and related extremality conditions in nonlinear programming.J. Optim. Theory Appl. 31 (1980), 2, 143–165. Zbl 0416.90062, MR 0600379, 10.1007/BF00934107 |
| Reference:
|
[10] Craven, B. D.: Invex functions and constrained local minima.Bull. Austral. Math. Soc. 24 (1981), 357–366. Zbl 0452.90066, MR 0647362, 10.1017/S0004972700004895 |
| Reference:
|
[11] Hanson, M. A.: On sufficiency of the Kuhn–Tucker conditions.J. Math. Anal. Appl. 80 (1981) 545–550. Zbl 0463.90080, MR 0614849, 10.1016/0022-247X(81)90123-2 |
| Reference:
|
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| Reference:
|
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| . |
