Optimization techniques are commonly applied to treatment design problems arising in radiation therapy. Previous literature on the application of operations research to radiotherapy has focused on small, site- specific treatment areas, e.g., prostate and head- and-neck sites. Tractability issues arise when these optimization methods are applied to larger treatment areas (such as total body irradiation) or higher- resolution treatments (such as stereotactic radiosurgery). We present a semi-infinite linear programming approach to solve high-resolution, convex quadratic optimization treatment problems in a reasonable amount of time. We also devise several computational improvements to the commonly used projected gradient algorithm that provide significant time savings when optimizations must be performed iteratively. Using our approaches, these previously unwieldy treatment planning problems can be solved in a clinically viable amount of time.