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Abstract. In this paper, we prove a necessary optimality theorem for a nonsmooth optimization problem in the face of data uncertainty, which is called a robust optimization problem. Recently, the robust optimization problems have been intensively studied by many authors. Moreover, we give examples showing that the convexity of the uncertain sets and the concavity of the constraint functions are essential in the optimality theorem. We present an example illustrating that our main assumptions in the optimality theorem can be weakened.

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In this paper, we consider the vector quasivariational inequalities and the vector quasivariational-like inequalities for multifunctions with vector values and prove some existence theorems of solutions for our inequalities. Also, we give the relationship between a kind of the vector variational inequality for multifunctions and a vector optimization problem involving nondifferentiable Lipschitz functions.

We define the generalized efficient solution which is more general than the weakly efficient solution for vector optimization problems, and prove the existence of the generalized efficient solution for nondifferentiable vector optimization problems by using vector variational-like inequalities for set-valued maps. ᮊ 1998 Academic Press

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We obtain a result on the Holder continuity of solutions to variational inequalities of the type previously studied by Noor, Mukherjee, and Verma. ᮊ 1997 Academic Press