Mechanisms of proton-proton inelastic cross-section growth in   multi-peripheral model within the framework of perturbation theory. Part 1
  We demonstrate a possibility of computation of inelastic scattering cross-section in a multi-peripheral model by application of the Laplace method to multidimensional integral over the domain of physical process. The constrained maximum point of scattering cross-section integral under condition of the energy-momentum conservation has been found. In the vicinity of this point the integrand is substituted for an expression of Gaussian type, which made possible to compute this integral numerically. The paper has two parts. The hunting procedure of the constrained maximum point is considered and the properties of this maximum point are discussed in the given part of the paper. It is shown that virtuality of all internal lines of the "comb" diagram reduced at the constrained maximum point with energy growth. In the second part of the paper we give some arguments in favor of consideration of the mechanism of virtuality reduction as the mechanism of the total hadron scattering cross-section growth, which is not taken into account within the framework of Regge theory. 
I. V. Sharf, G. O. Sokhrannyi, A. V. Tykhonov, K. V. Yatkin, N. A.   Podolyan, M. A. Deliyergiyev, V. D. Rusov