Ersatz function method for minimizing a finite-valued function on a compact set

A method is proposed for solving optimization problems with continuous variables and a function taking a large finite set of values. Problems of this type arise in the multicriteria construction of a control rule for a discrete-time dynamical system whose performance criteria coincide with the number of violations of requirements imposed on the system. The rule depends on a finite set of parameters whose set of admissible values defines a collection of admissible control rules. An example is the problem of choosing a control rule for a cascade of reservoirs. The optimization method is based on solving a modified problem in which the original function is replaced by a continuous ersatz function. A theorem on the relation between the average-minimal values of the original and ersatz functions is proved. Optimization problems are solved with power-law ersatz functions, and the influence exerted by the exponent on the quality of the solution is determined. It is experimentally shown that the solutions produced by the method are of fairly high quality.