The essence of invexity. (English) Zbl 0552.90077

Martin, D. H.

J. Optimization Theory Appl. 47, 65-76 (1985).

Summary: The notion of invexity was introduced into optimization theory by M. A. Hanson [J. Math. Anal. Appl. 80, 545–550 (1981; Zbl 0463.90080)] as a very broad generalization of convexity. A smooth mathematical program of the form \[ \text{minimize } f(x) \text{ subject to } g(x)\leq 0, \quad x\in {\mathcal D}\subseteq {\mathbb R}^ n, \] is invex if there exists a function \(\eta : {\mathcal D}\times {\mathcal D}\to {\mathbb R}^ n\) such that, for all \(x,u\in {\mathcal D}\), \[ f(x)-f(u)-f'(u)\eta (x,u)\geq 0,\text{ and }g(x)-g(u)-g'(u)\eta(x,u)\geq 0. \] The convex case corresponds of course to \(\eta(x,u)\equiv x-u\); but, as Hanson showed, invexity is sufficient to imply both weak duality and that the Kuhn-Tucker conditions are sufficient for global optimality. It is shown here that elementary relaxations of the conditions defining invexity lead to modified invexity notions which are both necessary and sufficient for weak duality and Kuhn-Tucker sufficiency.

MSC:

90C26	Nonconvex programming, global optimization
49K10	Optimality conditions for free problems in two or more independent variables
49M37	Numerical methods based on nonlinear programming
90C46	Optimality conditions and duality in mathematical programming
90C55	Methods of successive quadratic programming type
49N15	Duality theory (optimization)

Keywords:

invexity; weak duality; Kuhn-Tucker conditions; global optimality; relaxations

Citations:

Zbl 0463.90080

References:

[1]	Hanson, M. A.,On Sufficiency of the Kuhn-Tucker Conditions, Journal of Mathematical Analysis and Applications, Vol. 80, pp. 545-550, 1981. · Zbl 0463.90080
[2]	Craven, B. D.,Duality for Generalized Convex Fractional Programs, Generalized Concavity in Optimization and Economics, Edited by S. Schaible and W. T. Ziemba, Academic Press, New York, New York, 1981. · Zbl 0534.90089
[3]	Craven, B. D.,Invex Functions and Constrained Local Minima, Bulletin of the Australian Mathematical Society, Vol. 24, pp. 357-366, 1981. · Zbl 0452.90066
[4]	Hanson, M. A., andMond, B.,Further Generalizations of Convexity in Mathematical Programming, Journal of Information and Optimization Sciences, Vol. 3, pp. 25-32, 1982. · Zbl 0475.90069
[5]	Craven, B. D., andGlover, B. M.,Invex Functions and Duality, Journal of the Australian Mathematical Society, Series A (to appear). · Zbl 0565.90064
[6]	Hanson, M. A., andMond, B.,Necessary and Sufficient Conditions for Global Optimality and Constrained Optimization, Florida State University, Tallahassee, Florida, FSU Statistics Report No. M-600, 1981.
[7]	Mangasarian, O. L.,Nonlinear Programming, McGraw-Hill Book Company, New York, New York, 1969.