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 max(2) f 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. (C) 2002 Elsevier Science Ltd. All rights reserved.
global optimization under uncertainty double optimality