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In this chapter, we consider a class of multiobjective optimization problems with inequality, equality and vanishing constraints. For the scalar case, this class of problems reduces to the class of mathematical programs with vanishing constraints recently appeared in literature. We show that under fairly mild assumptions some constraint qualifications like Cottle constraint qualification, Slater constraint qualification, Mangasarian-Fromovitz constraint qualification, linear independence constraint qualification, linear objective constraint qualification and linear constraint qualification do not hold at an efficient solution, whereas the standard generalized Guignard constraint qualification is sometimes satisfied. We introduce suitable modifications of above mentioned constraint qualifications, establish relationships among them and derive the Karush-Kuhn-Tucker type necessary optimality conditions for efficiency.

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In this paper, we consider the multiobjective optimization problems involving the differentiable V-r-invex vector valued functions. Under the assumption of V-r-invexity, we use the Stampacchia type vector variational-like inequalities as tool to solve the vector optimization problems. We establish equivalence among the vector critical points, the weak efficient solutions and the solutions of the Stampacchia type weak vector variational-like inequality problems using Gordan?s separation theorem under the V-r-invexity assumptions. These conditions are more general than those appearing in the literature.