Bit error rate (BER) minimization and SNR-gap maximization, two robustness optimization problems, are solved, under average power and bitrate constraints, according to the waterfilling policy. Under peak power constraint the solutions differ and this paper gives bit-loading solutions of both robustness optimization problems over independent parallel channels. The study is based on analytical approach, using generalized Lagrangian relaxation tool, and on greedy-type algorithm approach. Tight BER expressions are used for square and rectangular quadrature amplitude modulations. Integer bit solution of analytical continuous bitrates is performed with a new generalized secant method. The asymptotic convergence of both robustness optimizations is proved for both analytical and algorithmic approaches. We also prove that, in the conventional margin maximization problem, the equivalence between SNR-gap maximization and power minimization does not hold with peak-power limitation. Based on a defined dissimilarity measure, bit-loading solutions are compared over Rayleigh fading channel for multicarrier systems. Simulation results confirm the asymptotic convergence of both resource allocation policies. In nonasymptotic regime the resource allocation policies can be interchanged depending on the robustness measure and on the operating point of the communication system. The low computational effort leads to a good trade-off between performance and complexity. [Read More]