Influencer marketing has become a thriving industry 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 that can create and publish posts of various types (e.g. text, image, video) for the promotion of a target product. The campaign's objective is to maximize across one or multiple online social platforms some impact metric of interest, e.g. number of impressions, sales (ROI), or audience reach. In this work, we present an original continuous formulation of the budgeted influencer marketing problem as a convex program. We further propose an efficient iterative algorithm based on the Frank-Wolfe method, that converges to the global optimum and has low computational complexity. We also suggest a simpler near-optimal rule of thumb, which can perform well in many practical scenarios. We test our algorithm and the heuristic against several alternatives from the optimization literature as well as standard seed selection methods and validate the superior performance of Frank-Wolfe in execution time and memory, as well as its capability to scale well for problems with very large number (millions) of social users.