Per my calculations, unless there are further restrictions on the parameters, that is non-convex. For example, consider the simplified two variable problem, and ignoring the linear term, which doesn’t affect convexity:
minimize(p1/(x-a*p1-b*p2) + p2/(x-a*p1-b*p2)) subject to p1 >= 0, p2 >= 0, p1 + p2 == 1.
The Hessian of the objective function with respect to p1 and p2 is indefinite when evaluated at x = 0.5, a = 0.4, b = 0.5, p1 = 0.5, p2 = 0.5. Therefore, the objective function is neither convex nor concave..
Please don’t ask for help implementing a problem in CVX until you have proven it is convex Why isn't CVX accepting my model? READ THIS FIRST! .