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This paper deals with a vector optimization problem with restric- tions of inequalities involving nondifferentiable maps in Banach spaces. Some new concepts of -E-semidifferentiablity, semi E-invexity and semi E-type-I maps in Banach spaces are introduced. A necessary optimality condition and a few sufficient optimality conditions are obtained by generalizing alternative theorem of Gordan type and using semi E-type-I maps. Moreover, weak, strong and converse duality results are proved under various types of semi E-type-I maps assumptions.

ON A NONDIFFERENTIABLE VECTOR OPTIMIZATION INVOLVING SEMI $E$-TYPE-I MAPS IN BANACH SPACES