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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
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 $
[1] Antczak, T.: An $\eta $-approximation approach to nonlinear mathematical programming involving invex functions. Numer. Funct. Anal. Optim. 25 (2004), 5–6, 423–438. DOI 10.1081/NFA-200042183 | MR 2106268
[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. DOI 10.1007/s10957-006-9069-9 | MR 2303801
[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. DOI 10.1080/01630560701190265 | MR 2302701 | Zbl 1141.90538
[4] Antczak, T.: A second order $\eta $-approximation method for constrained optimization problems involving second order invex functions. Appl. Math. 54 (2009), 433–445. DOI 10.1007/s10492-009-0028-2 | MR 2545410 | Zbl 1212.90307
[5] Bazaraa, M. S., Sherali, H. D., Shetty, C. M.: Nonlinear Programming: Theory and Algorithms. John Wiley and Sons, New York 1991. MR 2218478
[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.
[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. MR 0885768 | Zbl 0622.90068
[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.
[9] Ben-Tal, A.: Second-order and related extremality conditions in nonlinear programming. J. Optim. Theory Appl. 31 (1980), 2, 143–165. DOI 10.1007/BF00934107 | MR 0600379 | Zbl 0416.90062
[10] Craven, B. D.: Invex functions and constrained local minima. Bull. Austral. Math. Soc. 24 (1981), 357–366. DOI 10.1017/S0004972700004895 | MR 0647362 | Zbl 0452.90066
[11] Hanson, M. A.: On sufficiency of the Kuhn–Tucker conditions. J. Math. Anal. Appl. 80 (1981) 545–550. DOI 10.1016/0022-247X(81)90123-2 | MR 0614849 | Zbl 0463.90080
[12] Mangasarian, O. L.: Nonlinear Programming. McGraw-Hill, New York 1969. MR 0252038 | Zbl 0194.20201
[13] Rockafellar, R. T.: Convex Analysis. Princeton University Press, 1970. MR 0274683 | Zbl 0193.18401
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