The accuracy of intensity modulated proton therapy (IMPT) is sensitive to range uncertainties. Geometric margins, as dosimetric surrogates, are ineffective and robust optimization strategies are needed. These, however, lead to increased normal tissue dose. We explore here how this dose increase can be reduced by increasing the maximum tumor dose instead. We focus on range uncertainties, modeled by scaling the stopping powers 5% up (undershoot) or down (overshoot) compared to the nominal scenario. Robust optimization optimizes for target dose conformity in the most likely scenario, not the worst, while constraining target coverage for the worst-case scenario. Non-robust plans are also generated. Different maximum target doses are applied (105% versus 120% versus 140%) to investigate the effect on normal tissue dose reduction. The method is tested on a homogeneous and a lung phantom and on a liver patient. Target D99 of the robust plans equals the prescription dose of 60 GyEWe use the symbol GyE for the correct notation of Gy(RBE). for all scenarios, but decreases to 36 GyE for the non-robust plans. The mean normal tissue dose in a 2 cm ring around the target is 11% to 31% higher for the robust plans. This increase can be reduced to -8% and 3% (compared to the non-robust plan) by allowing a maximum tumor dose of 120% instead of 105%. Thus robustness leads to more normal tissue dose, but it can be compensated by allowing a higher maximum tumor dose.