The essence of invexity.(English)Zbl 0552.90077

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)

Zbl 0463.90080
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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.
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