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Optimality conditions and duality for semi-infinite programming involving B-arcwise connected functions. (English) Zbl 1187.90302

By relaxing the definitions of arcwise connected functions and B-vex functions, the paper introduces the generalized convexity concept of (strictly) B-arcwise connected functions. Fritz John and Karush-Kuhn-Tucker type necessary and sufficient optimality conditions as well as duality results are given for semi-infinite optimization problems involving such functions.

MSC:

90C34 Semi-infinite programming
90C46 Optimality conditions and duality in mathematical programming
26B25 Convexity of real functions of several variables, generalizations
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[1] Avriel M.: Nonlinear Programming: Analysis and Methods. Prentice Hall, Englewood Cliffs, New Jersey (1976) · Zbl 0361.90035
[2] Avriel M., Zang I.: Generalized arcwise-connected functions and characterizations of local-global minimum properties. J. Optim. Theory Appl. 32, 407–425 (1980) · Zbl 0422.90069
[3] Bector C.R., Singh C.: B-Vex functions. J. Optim. Theory Appl. 70, 237–253 (1991) · Zbl 0793.90069
[4] Bhatia D., Mehra A.: Optimality and duality involving arcwise connected and generalized arcwise connected functions. J. Optim. Theory Appl. 100, 181–194 (1999) · Zbl 0922.90124
[5] Borwein J.M.: Direct theorems in semi-infinite convex programming. Math. Prog. 21, 301–318 (1981) · Zbl 0469.90065
[6] Charnes A., Cooper W.W., Kortanet K.O.: Duality in semi-infinite programs and some works of Haar and Caratheodory. Manag. Sci. 9, 209–228 (1963) · Zbl 0995.90615
[7] Davar S., Mehra A.: Optimality and duality for fractional programming problems involving arcwise connected functions and their generalizations. J. Math. Anal. Appl. 263, 666–682 (2001) · Zbl 1109.90334
[8] Goberna M.A., López M.A.: On duality in semi-infinite programming and existence theorems for linear inequalities. J. Math. Anal. Appl. 230, 173–192 (1999) · Zbl 0916.90255
[9] Hettich R., Kortanet K.O.: Semi-infinite programming: theory, methods, and applications. SIAM Rev. 35, 380–429 (1993) · Zbl 0784.90090
[10] Jeroslow R.G.: Uniform duality in semi-infinite convex optimization. Math. Prog. 27, 144–154 (1983) · Zbl 0556.49008
[11] Karney D.F.: Duality gaps in semi-infinite linear programming and approximation problem. Math. Prog. 20, 129–143 (1981) · Zbl 0448.90037
[12] Karney D.F., Morley T.D.: Limiting Lagrangians: a primal approach. J. Optim. Theory Appl. 48, 163–174 (1986) · Zbl 0556.90069
[13] Li S.Z.: On dual gap of semi-infinite programming. Acta Mathematica Scientia 20, 1–5 (2000) (in Chinese) · Zbl 0973.90081
[14] Liu A.L.: The duality of semi-infinite linear programs. J. Eest China Normal Univ. (Natural Seience) 1, 7–12 (1988) (in Chinese) · Zbl 0679.90034
[15] López M.A., Vercher E.: Optimality conditions for nondifferentiable convex semi-infinite programming. Math. Prog. 27, 307–319 (1983) · Zbl 0527.49029
[16] Mehra A., Bhatia D.: Optimality and duality for minmax problems involving arcwise connected and generalized arcwise connected functions. J. Math. Anal. Appl. 231, 425–445 (1999) · Zbl 0916.90245
[17] Polak E.: On the mathematical foundations of nondifferentiable optimization in engineering design. SIAM Rev. 29, 21–89 (1987)
[18] Reemtsen R., Rückmann J.J.: Semi-infinite Programming. Kluwer Publishers, Boston (1998) · Zbl 0908.90255
[19] Rückmann J.J., Shapiro A.: First-order optimality conditions in generalized semi-infinite programming. J. Optim. Theory Appl. 101, 677–691 (1999) · Zbl 0956.90055
[20] Singh C.: Elementary properties of arcwise connected set and functions. J. Optim. Theory Appl. 41, 85–103 (1990)
[21] Zhang Q.X.: Optimality conditions and duality for arcwise semi-infinite programming with parametric inequality constraints. J. Math. Anal. Appl. 196, 998–1007 (1995) · Zbl 0842.90114
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