A non-convex problem of mathematical programming is considered, which permissible region is a simplex. A two-stage algorithm is proposed for approximate solution of the problem. The region of global optimum is determined using the Ψ-transform method at the first stage; local “fine-tuning” of the solution is performed at the second stage. The Ψ-transform was modified taking into account the special features of the problem under consideration. Ψ-function is determined according to the results of statistical tests implemented using the generator of random points uniformly distributed over the simplex. The proposed method of reflection of regular simplexes is used for fine-tuning of the solution. An example of application of the developed algorithm for solving the problem of optimization of component composition of the hydrocarbon mixture is presented.