## 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)

Zbl 0475.90069