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**Strong and weak invexity in mathematical programming.**
*(English)*
Zbl 0566.90086

For a primal differentiable nonlinear programming problem satisfying a weakened convex property now called invex, M. A. Hanson and B. Mond [J. Inf. Optimization Sci. 3, 25-32 (1982; Zbl 0475.90069)] and others showed that Kuhn-Tucker conditions are sufficient for a global minimum, and duality holds between the primal problem and its formal Wolfe dual. The invex property is now generalized to \(\rho\)-invex, in which the defining inequality for invex holds approximately, to within a term depending on a parameter \(\rho\) which may be positive (strongly invex) or negative (weakly invex). This also generalizes Vial’s \(\rho\)- convex. Several sufficient conditions are obtained for a function to be \(\rho\)-invex. Kuhn-Tucker sufficiency results and duality results are obtained using \(\rho\)-invex functions, extending recent results for invex functions.

### MSC:

90C30 | Nonlinear programming |

49M37 | Numerical methods based on nonlinear programming |

90C25 | Convex programming |

90C55 | Methods of successive quadratic programming type |

49N15 | Duality theory (optimization) |