Characterizations of the solution sets of pseudoinvex programs and variational inequalities

Springer Science and Business Media LLC - 2011
Caiping Liu1, Xinmin Yang2, Heung Wing Joseph Lee3
1College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China
2Department of Mathematics, Chongqing Normal University, Chongqing, 400047, China
3Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong,China

Tóm tắt

Abstract A new concept of nondifferentiable pseudoinvex functions is introduced. Based on the basic properties of this class of pseudoinvex functions, several new and simple characterizations of the solution sets for nondifferentiable pseudoinvex programs are given. Our results are extension and improvement of some results obtained by Mangasarian (Oper. Res. Lett., 7, 21-26, 1988), Jeyakumar and Yang (J. Optim. Theory Appl., 87, 747-755, 1995), Ansari et al. (Riv. Mat. Sci. Econ. Soc., 22, 31-39, 1999), Yang (J. Optim. Theory Appl., 140, 537-542, 2009). The concepts of Stampacchia-type variational-like inequalities and Minty-type variational-like inequalities, defined by upper Dini directional derivative, are introduced. The relationships between the variational-like inequalities and the nondifferentiable pseudoinvex optimization problems are established. And, the characterizations of the solution sets for the Stampacchia-type variational-like inequalities and Minty-type variational-like inequalities are derived.

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