Mathematical models of process of carry of molecules are developed the channel, allowing to optimize a flowing part hybrid turbomolecular the vacuum pump (TMP) in a wide range speeds. Interaction of steps and flowing is investigated parts turbomolecular the vacuum pump with various constructive schemes. The design procedure of parameters is developed ТМР with axial-radial a stream of gas. The algorithm is developed optimization flowing part ТМР with axial and axial-radial stream gas, laws of change of geometrical parameters are received on to steps of a compound flowing part. Theoretical researches were carried out with use of a method of statistical tests and a method of angular factors. Experimental researches were carried out in laboratory conditions on models and experimental samples. Algorithm of optimization at presence nonlinear functional restrictions on controlled parameters uses algorithm sliding the admission and a method of absolute penal functions under condition of maintenance set speed of action of the pump on the chosen gas, and also on several gases. The problem of optimization is put as variational. Therefore the absolute penalty function is used, if in current optimization step S@sub real@ will be less than S: F(x,@rho@) =@Phi@(x)+@rho@ |(S@sub real@ - S) < 0| , where @rho@- penalty parameter, the upper value of which is restricted by bad dependence F(x,@rho@). After calculation of evacuation characteristic at current values of controlled parameters it is possible the situation, when P is less, than the limit rest pressure P@sub min@. In this case the value S@sub real@ is taken 0, and the target function is formed as follows: F'(x, @rho@, @rho@ @sub 1@)=Ф(x)+@rho@ |(S@sub real@ - S) < 0 |+ @rho@ @sub 1@ |{(lg (P)- lg(P@sub min@)}< 0| , @rho@ is chosen equal 10@super 5@. Minimization of flow part volume of TMP with axial and, the more, with axial-radial gas flow with given technical conditions for nitrogen (N@sub 2@) does not guarantee the required operation rate for light gases (H@sub 2@, He) due to nonlinear character of dependence S@sub max@ of axial and axial-radial rotors on round velocity when U/V <1. Therefore, if, for example, it is needed to minimize the flow part's volume of a TMP with given evacuation rate S at P for nitrogen N@sub 2@ and given evacuation rate S at P for hydrogen H@sub 2@, the absolute penalty function is formed as follows: F''(x, @rho@, @rho@ @sub 1@ , @rho@ @sub 2@ , @rho@ @sub 3@)= Ф(x)+@rho@|(S@sub real@ - S) < 0|+ @rho@ @sub 1@|{lg(P)-lg(P@sub min@)} < 0|+ @rho@ @sub 2@ |(S@sub real@, Н- S) < 0|+@rho@ @sub 3@ |{lg (P) - lg(P@sub min@, Н)} <0| , where P@sub min@, H - limit rest pressure for H@sub 2@. Naturally, it is necessary to calculate the discrete evacuation characteristic for H@sub 2@.Often in technical conditions are included different restrictions on other outgoing and optimization parameters, e.g. rotation frequency f, external diameter D@sub 2@ etc.