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

Jeyakumar, V.

Methods Oper. Res. 55, 109-125 (1985).

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)

Keywords:

rho-invexity; rho-convexity; invexity conditions; strong and weak invexity; differentiable nonlinear programming; Kuhn-Tucker sufficiency results; duality results

Citations:

Zbl 0475.90069