The problem of optimizing traffic on the road network is considered. Congestions within road networks continue to be relevant, and in situations where there is no systematic approach to its solution, losses due to the inefficiency of the infrastructure can be very large. Three modified modern algorithms for solving variational inequalities applied to the user-optimal traffic equilibrium search problem. For that, mathematical model of traffic equilibrium corresponding to the first principle of Wardrop is provided, with formulation in form of variational inequality. The algorithms are adaptive versions of extragradient algorithm, extrapolation from the past, and Malitsky algorithm. We do not need to know Lipschitz constant of operator to select the step size. All three algorithms are extended with step size modification rule which allows both step increasing and decreasing. Numerical experiments are conducted with the toy problem (Braess classical example) and with the well-known Sioux Falls transportation network. Developed software complex for applying algorithms based on variational inequalities allows to make empirical conclusions about behavior of algorithms during calculation process.