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**Optimality criteria in mathematical programming involving generalized invexity.**
*(English)*
Zbl 0647.90076

Constrained optimization problems of the form (1) minimize f(x) subject to \(x\in X\subseteq R^ n\), g(x)\(\leq 0\), with differentiable functions f, g f type I or type II are considered: The functions f, g are called of type I with respect to a vector function \(\eta\) (x) at \(x_ 0\) if the relations
\[
f(x)-f(x_ 0)\geq [\nabla_ xf(x_ o)]' \eta (x),\quad - g(x_ 0)\geq [\nabla_ xg(x_ o)] \eta (x)
\]
hold for all feasible x of the problem (1).

Similarly f, g are called of type II with respect to x at \(x_ 0\), if \[ f(x_ 0)-f(x)\geq [\nabla_ xf(x)]' \eta (x),\quad and\quad -g(x)\geq \nabla_ xg(x) \eta (x) \] are satisfied for all feasible solutions of the problem (1). Various sufficient conditions, under which the functions f, g are of type I or II are given. Sufficient optimality conditions for the problem (1), in which f, g are of type I or II are proved.

Similarly f, g are called of type II with respect to x at \(x_ 0\), if \[ f(x_ 0)-f(x)\geq [\nabla_ xf(x)]' \eta (x),\quad and\quad -g(x)\geq \nabla_ xg(x) \eta (x) \] are satisfied for all feasible solutions of the problem (1). Various sufficient conditions, under which the functions f, g are of type I or II are given. Sufficient optimality conditions for the problem (1), in which f, g are of type I or II are proved.

Reviewer: K.Zimmermann

### MSC:

90C30 | Nonlinear programming |

49K05 | Optimality conditions for free problems in one independent variable |

### Keywords:

Sufficient optimality conditions
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\textit{N. G. Rueda} and \textit{M. A. Hanson}, J. Math. Anal. Appl. 130, No. 2, 375--385 (1988; Zbl 0647.90076)

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### References:

[1] | A. Ben-Israel and B. Mond; A. Ben-Israel and B. Mond · Zbl 0603.90119 |

[2] | Hanson, M. A., On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl., 80, 545-550 (1981) · Zbl 0463.90080 |

[3] | Hanson, M. A.; Mond, B., Necessary and Sufficient Conditions in Constrained Optimization, (FSU Statistics Report M 683 (1984), Florida State University, Department of Statistics: Florida State University, Department of Statistics Tallahassee, Florida) · Zbl 0622.49005 |

[4] | Kaul, R. N.; Kaur, S., Optimality criteria in nonlinear programming involving non-convex functions, J. Math. Anal. Appl., 105, 104-112 (1985) · Zbl 0553.90086 |

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