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.


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