The deliverability of proton pencil beam scanning (PBS) plans is subject to the minimum monitor-unit (MU) constraint, while the delivery efficiency depends on the number of proton energy layers. This work develops an inverse optimization method for generating efficiently deliverable PBS plans. The proposed minimum-MU and sparse-energy-layer (MMSEL) constrained inverse optimization method utilizes iterative convex relaxations to handle the nonconvexity from minimum-MU constraint and dose-volume constraints, and regularizes group sparsity of proton spots to minimize the number of energy layers. The tradeoff between plan quality and delivery efficiency (in terms of the number of used energy layers) is controlled by the objective weighting of group sparsity regularization. MMSEL consists of two steps: first minimize for appropriate energy layers, and then with selected energy layers solve for the deliverable PBS plan. The solution algorithm for MMSEL is developed using alternating direction method of multipliers (ADMM). Range and setup uncertainties are modelled by robust optimization. MMSEL was validated using representative prostate, lung, and head-and-neck (HN) cases. The minimum-MU constraint was strictly enforced for all cases, so that all plans were deliverable. The number of energy layers was reduced by MMSEL to 78%, 76%, and 61% for prostate, lung and HN, respectively, while the similar plan quality was achieved. The number of energy layers was reduced by MMSEL to 54%, 57%, and 37% for prostate, lung and HN, respectively, while the plan quality was comprised and acceptable. MMSEL is proposed to strictly enforce minimum-MU constraint and minimize the number of energy layers during inverse optimization for efficiently deliverable PBS plans. In particular, the preliminary results suggest MMSEL potentially enables 25% to 40% reduction of energy layers while maintaining the similar plan quality.