Minimization of A Nondifferentiable Convex Function Defined not Everywhere

Generalized Level Method with Approximated Data

“…. , m, x 2 X, y 2 Y exactly as program (1). As in decomposition, it is quite natural that the problem data of (1) are themselves given as optimal values of inner programs (2).…”

“…LEMMA 3 [1] If for a nonincreasing nonnegative sequence fy j g there exist constants 0 < c 1 < 1 and c 2 > 0 such that rþs ! c 1 r for some r !…”

“…Nevertheless, the partition into the index sets J 1 and J 2 remains unchanged except for the current index. The generalization of the original level method as described in [2] is done in two ways: firstly, another strategy to handle constraints hðyÞ 0 is used (the same as done in [1] to allow the function ' to be finite not everywhere on Y); secondly, we allow the oracle to deliver approximated data only, as will be the case in most applications. However, we suppose to be able to define the accuracy in advance.…”