Influencer marketing has become in the recent years a thriving industry that includes more than 1120 agencies worldwide and with a global market value expected to reach 15 billion dollars by 2022. The advertising problem that such agencies face is the following: given a monetary budget find a set of appropriate influencers on a social platform and recruit them to create a number of posts for the promotion of a certain product. The objective of the campaign is to maximize some impact metric, e.g. the number of impressions, the sales, or the audience reach. In this work, we present an original formulation of the budgeted campaign orchestration problem as a convex program, and further derive a near-optimal algorithm to solve it efficiently. The proposed algorithm has low computational complexity and can scale well for problems with large numbers (millions) of social users, encountered in real-world platforms. We apply our algorithm to a Twitter data set and illustrate the optimal campaign performance for various metrics of interest.