Intensity-modulated radiation therapy (IMRT) is a state-of-the-art technique for administering radiation to cancer patients. The goal of a treatment is to maximize the radiation absorbed by the tumor and minimize that absorbed by the surrounding critical organs. Since a plan can almost never be found that both kills the tumor and completely avoids irradiating critical organs, the medical physics community has quantified the sacrifices that can be tolerated in so-called dose-volume constraints. Efficiently imposing such constraints, which are fundamentally combinatorial in nature, poses a major challenge due to the large amount of data. Also, the IMRT technology limits which dose distributions are actually deliverable. So, we seek a physically deliverable dose distribution that at the same time meets the minimum tumor dose prescription and satisfies the dose-volume constraints. We propose a new greedy algorithm and show that it converges to a local minimum of the stated formulation of the fluence map problem. Numerical comparison is made to an approach representative of the leading commercial software for IMRT planning. We find our method produces plans of competitive quality with a notable improvement in computational performance. Our efficiency gain is most aptly attributed to a new interior-point gradient algorithm for solving the nonnegative least squares subproblem every iteration. Convergence is proven and numerical comparisons are made to other leading methods demonstrating this solver is well-suited for the subproblem.