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mathematical programming; second order $\eta $-approximated optimization problem; second order invex function; second order optimality conditions
A new approach for obtaining the second order sufficient conditions for nonlinear mathematical programming problems which makes use of second order derivative is presented. In the so-called second order $\eta $-approximation method, an optimization problem associated with the original nonlinear programming problem is constructed that involves a second order $\eta $-approximation of both the objective function and the constraint function constituting the original problem. The equivalence between the nonlinear original mathematical programming problem and its associated second order $\eta $-approximated optimization problem is established under second order invexity assumption imposed on the functions constituting the original optimization problem.
[1] Antczak, T.: An $\eta $-approximation approach to nonlinear mathematical programming problems involving invex functions. Numer. Funct. Anal. Optimization 25 (2004), 423-438. DOI 10.1081/NFA-200042183 | MR 2106268
[2] Bazaraa, M. S., Sherali, H. D., Shetty, C. M.: Nonlinear Programming. Theory and Algorithms. John Wiley & Sons New York (1993). MR 2218478 | Zbl 0774.90075
[3] Bector, C. R., Bector, B. K.: (Generalized)-bonvex functions and second order duality for a nonlinear programming problem. Congr. Numerantium 52 (1985), 37-52.
[4] Bector, C. R., Bector, M. K.: On various duality theorems for second order duality in nonlinear programming. Cah. Cent. Etud. Rech. Opér. 28 (1986), 283-292. MR 0885768 | Zbl 0622.90068
[5] Bector, C. R., Chandra, S.: Generalized bonvex functions and second order duality in mathematical programming. Res. Rep. 85-2 Department of Actuarial and Management Sciences, University of Manitoba Winnipeg (1985).
[6] Bector, C. R., Chandra, S.: (Generalized) bonvexity and higher order duality for fractional programming. Opsearch 24 (1987), 143-154. MR 0918321 | Zbl 0638.90095
[7] Ben-Israel, A., Mond, B.: What is invexity?. J. Aust. Math. Soc. Ser. B 28 (1986), 1-9. DOI 10.1017/S0334270000005142 | MR 0846778 | Zbl 0603.90119
[8] Ben-Tal, A.: Second-order and related extremality conditions in nonlinear programming. J. Optimization Theory Appl. 31 (1980), 143-165. DOI 10.1007/BF00934107 | MR 0600379 | Zbl 0416.90062
[9] Craven, B. D.: Invex functions and constrained local minima. Bull. Aust. Math. Soc. 24 (1981), 357-366. DOI 10.1017/S0004972700004895 | MR 0647362 | Zbl 0452.90066
[10] 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
[11] Mangasarian, O. L.: Nonlinear Programming. McGraw-Hill New York (1969). MR 0252038 | Zbl 0194.20201
[12] Martin, D. H.: The essence of invexity. J. Optimization Theory Appl. 47 (1985), 65-76. DOI 10.1007/BF00941316 | MR 0802390 | Zbl 0552.90077
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