Characterizations of the solution sets of pseudoinvex programs and variational inequalities
Tóm tắt
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.
Từ khóa
Tài liệu tham khảo
Mangasarian OL: A simple characterization of solution sets of convex programs. Oper Res Lett 1988, 7: 21–26. 10.1016/0167-6377(88)90047-8
Burke JV, Ferris MC: Characterization of solution sets of convex programs. Oper Res Lett 1991, 10: 57–60. 10.1016/0167-6377(91)90087-6
Jeyakumar V: Infinite -dimensional convex programming with applications to constrained approximation. J Optim Theory Appl 1992, 75: 469–586.
Jeyakumar V, Yang XQ: Convex composite multi-objective nonlinear programming. Math Program 1993, 59: 325–343. 10.1007/BF01581251
Jeyakumar V, Yang XQ: On characterizing the solution sets of pseudolinear programs. J Optim Theory Appl 1995, 87: 747–755. 10.1007/BF02192142
Ansari QH, Schaible S, Yao JC: η -pseudolinearity. Riv Mat Sci Econ Soc 1999, 22: 31–39.
Yang XM: On characterizing the solution sets of pseudoinvex extremum problems. J Optim Theory Appl 2009, 140: 537–542. 10.1007/s10957-008-9470-7
Mancino OG, Stampacchia G: Convex programming and variational inequalities. J Optim Theory Appl 1972, 9: 3–23. 10.1007/BF00932801
Parida J, Sahoo M, Kumar A: A variational-like inequality problem. B Aust Math Soc 1989, 39: 225–231. 10.1017/S0004972700002690
Crespi GP, Ginchev I, Rocca M: Minty variational inequalities, increase along rays property and optimization. J Optim Theory Appl 2004, 123: 479–496. 10.1007/s10957-004-5719-y
Crespi GP, Ginchev I, Rocca M: Existence of solutions and star-shapedness in minty variational inequalities. J Glob Optim 2005, 32: 485–494. 10.1007/s10898-003-2685-0
Ivanov VI: Optimization and variational inequalities with pseudoconvex functions. J Optim Theory Appl 2010, 146: 602–616. 10.1007/s10957-010-9682-5
Weir T, Mond B: Preinvex functions in multiple objective optimization. J Math Anal Appl 1988, 136: 29–38. 10.1016/0022-247X(88)90113-8
Mohan SR, Neogy SK.: On invex sets and preinvex functions. J Math Anal Appl 1995, 189: 901–908. 10.1006/jmaa.1995.1057
Yang XM, Yang XQ, Teo KL: Characterizations and applications of prequasi-Invex functions. J Optim Theory Appl 2001, 110: 645–668. 10.1023/A:1017544513305
Yang XM, Yang XQ, Teo KL: Generalized invexity and generalized invariant monotonicity. J Optim Theory Appl 2003, 117: 607–625. 10.1023/A:1023953823177
Sach PH, Penot JP: Charaterizations of generalized convexities via generalized directional derivative. Numer Funct Anal Optim 1998, 19: 615–634. 10.1080/01630569808816849
Diewert WE: Alternative charaterizations of six kinds of quasiconcavity in the nondifferentiable case with applications to nonsmooth programming. In Generalized Concavity in Optimization and Economics. Edited by: Schaible S, Ziemba WT. Academic Press, New York; 1981:51–93.