We present a generalization of the Uniform Boundedness Principle valid for random multivalued linear operators, i.e., multivalued linear operators taking values in the space L 0 ( , Y ) of random variables defined on a probability space ( , A, P ) with values in the Banach space Y . Namely, for a family of such operators that are continuous with positive probability, if the family is pointwise bounded with probability at least δ > 0, then the operators are uniformly bounded with a probability that in each case can be estimated in terms of δ and the index of continuity of the operator. To achieve this result, we develop the fundamental theory of multivalued linear operators on general topological vector spaces. In particular, we exhibit versions of the Closed Graph Theorem, the Open Mapping Theorem, and the Uniform Boundedness Principle for multivalued operators between F -spaces. (2000): 46A16, 46A30, 47A06, 47B80, 60H25. Mathematics Subject Classifications

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Abstract. Convex non linear optimization problems may not have a solution in innite dimension spaces. The aim of this paper is to formulate some new results in this topic by using "technical regularization" of the objective function. The rst result shows that a non linear convex proper lower semi-continuous function, on a Banach space which have the Radon-Nikodym property, could be minimized by using a small regularization. while the second one shows that this regularization can be chosen as small as required. In addition, application tracks are presented and illustrated by elementary examples.