Global optimization in problems with uncertainties: The gamma algorithm

Problems with uncertainties can be viewed and formalized making use of multifunctions or general set-valued functions. A new concept of global optimality is proposed which allows us to solve global optimization problems with uncertainties, in natural setting without imposing artificial constraints on uncertainties, nor introducing a kind of partial ordering (in order, to apply conventional optimality concepts and optimization techniques) nor considering solution ‘in probability’. With the new concept, deterministic optimization requires two optimization procedures. A study of the subject is presented with many illustrative examples. Then, a monotonic iterative algorithm is developed which renders approximate solutions with precision specified in advance. A notion of piecewise continuous function of several variables is proposed and the method is then generalized for uncertain functions defined by a closed set-valued function with piecewise continuous upper and lower boundaries. The max² ƒ reduction, precision and decomposition lemmas are proved. To facilitate practical applications, deferred deletions of sets with discontinuities are introduced, and convergence theorem is proved for the modified algorithm.