Gradient-based optimization is used to maximize the propulsive efficiency of a heaving and pitching flexible panel. Random initial conditions are sequentially improved until a minimum step size is reached, at which point the condition is considered locally optimal. Optimum pitch and heave motions are found to produce nearly twice the efficiencies of optimum heave-only motions. Particle Image Velocimetry (PIV) is used to investigate the flow structures at optimal conditions. Efficiency is globally optimized when (1) the Strouhal number is within an optimal range that varies weakly with amplitude and boundary conditions; (2) the panel is actuated at a resonant frequency of the fluidpanel system; (3) heave amplitude is tuned such that trailing edge amplitude is maximized while the flow along the body remains attached; and (4) the maximum pitch angle and phase lag are chosen so that the effective angle of attack is minimized. The multi-dimensionality and multimodality of the efficiency response demonstrate that experimental optimization appears to be well-suited for the design of flexible underwater propulsors.