Arc proton therapy (ArcPT) is an emerging modality in cancer treatments. It delivers the proton beams following a sequence of irradiation angles while the gantry is continuously rotating around the patient. Compared to conventional proton treatments (intensity modulated proton therapy, IMPT), the number of beams is significantly increased bringing new degrees of freedom that leads to potentially better cancer care. However, the optimization of such treatment plans becomes more complex and several alternative statements of the problem can be considered and compared in order to solve the ArcPT problem. Three such problem statements, distinct in their mathematical formulation and properties, are investigated and applied to solving the ArcPT optimization problem. They make use of (i) fast iterative shrinkage-thresholding algorithm (FISTA), (ii) local search (LS) and (iii) mixed-integer programming (MIP). The treatment plans obtained with those methods are compared among them, but also with IMPT and an existing state-of-the-art method: Spot-Scanning Proton Arc (SPArc). MIP stands out at low scale problems both in terms of dose quality and time delivery efficiency. FISTA shows high dose quality but experiences difficulty to optimize the energy sequence while LS is mostly the antagonist. This detailed study describes independent approaches to solve the ArcPT problem and depending on the clinical case, one should be cautiously picked rather than the other. This paper gives the first formal definition of the problem at stake, as well as a first reference benchmark. Finally, empirical conclusions are drawn, based on realistic assumptions.