Sector duration optimization (SDO) is a problem arising in treatment planning for stereotactic radiosurgery on Gamma Knife. Given a set of isocenter locations, SDO aims to select collimator size configurations and irradiation times thereof such that target tissues receive prescribed doses in a reasonable amount of treatment time, while healthy tissues nearby are spared. We present a multiobjective linear programming model for SDO with a state-of-the-art delivery system, Leksell Gamma Knife Icon, to generate a diverse collection of solutions so that one that best suits patient-specific needs can be chosen by clinicians. We develop a generic two-phase solution strategy based on the epsilon-constraint method for solving multiobjective optimization models, which aims to systematically increase the number of high-quality solutions obtained, instead of conducting a traditional uniform search. To improve solution quality further and accelerate the procedure, we incorporate some general and problem-specific enhancements. Moreover, we propose an alternative version of our two-phase strategy, which makes use of machine learning tools to devote the computational effort rather to solving epsilon-constraint models that are predicted to yield clinically desirable solutions. In our computational study on eight previously treated real test cases, a significant portion of obtained solutions outperforms clinical results and those from a single-objective model from the literature. In addition to significant benefits of the algorithmic enhancements, our experiments illustrate the usefulness of machine learning strategies, reducing the overall run times nearly by half while still capturing a sufficient selection of clinically desirable solutions.