  Many social phenomena are triggered by public opinion that is formed in the
process of opinion exchange among individuals. To date, from the engineering
point of view, a large body of work has been devoted to studying how to
manipulate individual opinions so as to guide public opinion towards the
desired state. Recently, Abebe et al. (KDD 2018) have initiated the study of
the impact of interventions at the level of susceptibility rather than the
interventions that directly modify individual opinions themselves. For the
model, Chan et al. (The Web Conference 2019) designed a local search algorithm
to find an optimal solution in polynomial time. However, it can be seen that
the solution obtained by solving the above model might not be implemented in
real-world scenarios. In fact, as we do not consider the amount of changes of
the susceptibility, it would be too costly to change the susceptibility values
for agents based on the solution.
  In this paper, we study an opinion optimization model that is able to limit
the amount of changes of the susceptibility in various forms. First we
introduce a novel opinion optimization model, where the initial susceptibility
values are given as additional input and the feasible region is defined using
the $\ell_p$-ball centered at the initial susceptibility vector. For the
proposed model, we design a projected gradient method that is applicable to the
case where there are millions of agents. Finally we conduct thorough
experiments using a variety of real-world social networks and demonstrate that
the proposed algorithm outperforms baseline methods.
