Stability of uncertain optimization problems

The problem of optimization of not completely defined functions, i.e. functions with parameters set only up to interval is considered. A review of existing approaches to solving optimization problems of incompletely specified functions with different kinds of uncertainty is given. The mathematical formulation of the problem of optimizing the functions with interval parameters and the method of its solution by reducing it to two optimization problems of completely defined functions i.e. functions with exactly known parameters (the determination method) is described. It is shown that the solution of the optimization problem of not fully defined functions also requires consideration of the problem of determining the optimum stability to variations in the values of the function parameters. In this regard, we introduce notions of macrostability and microstability of optimization problem of fully defined functions. Necessary and sufficient conditions for the macrostability of optimization problem of completely specified functions are given. An algorithm for checking of macrostability is presented. An example of checking macrostability of particular problem with this algorithm (assignment problem) is given. We also present an algorithm for checking microstability of optimization problem of fully defined function. To solve such problems methods of interval mathematics are used.