Argos satellite telemetry is an important tool for studying marine animal movement, but the resulting locations are often irregularly spaced and contain high measurement uncertainty, making trajectory reconstruction difficult. This study evaluated whether convex optimisation could reconstruct plausible tiger shark (Galeocerdo cuvier) trajectories from noisy Argos data and compared its performance with standard state-space model (SSM) baselines. Using telemetry data from sharks tracked in Shark Bay, Western Australia, trajectories were reconstructed on 6-hour and 12-hour temporal grids using two convex formulations; an uncertainty-weighted quadratic model and a Huber penalty variant, both combined with a smoothness penalty and maximum-speed constraint. These convex formulations were compared with random walk, correlated random walk, and move-persistence SSM baselines. Performance across 46 trajectory segments was assessed using a whitened sum of squared error criterion based on Argos uncertainty ellipses. Under this criterion, the 6-hour convex quadratic model performed best overall, with the lowest mean rank across all trajectory segments and the highest number of first-place performances among the models compared. The results suggest that convex optimisation provides a useful and biologically interpretable alternative for trajectory reconstruction from noisy telemetry data when performance is assessed by uncertainty-weighted agreement with Argos observations.