Low Earth Orbit (LEO) satellite Doppler measurements offer an alternative source of positioning information for Internet of Things devices when Global Navigation Satellite System signals are unavailable. Conventional local optimisation methods depend on an accurate initial position and can fail during a complete cold start. We propose an initialization-free LEO Doppler positioning framework based on semidefinite relaxation. The method reformulates the nonlinear positioning problem into a semidefinite programme to obtain a receiver position estimate without prior location knowledge. We analyse relaxation tightness under noiseless conditions and derive sufficient noise bounds under parameter disturbances. Simulations and field experiments using opportunistic Iridium-NEXT signals evaluate initialization sensitivity, positioning accuracy and computational cost. In the field experiment, the proposed method achieves a three-dimensional positioning error of 140 m without an initial position, whereas local solvers converge to poor solutions under large initial offsets. Using the semidefinite estimate to initialise Gauss-Newton or Dog-Leg reduces the positioning error to 130 m. The framework provides a cold-start initialization stage for subsequent local refinement in opportunistic LEO positioning. The implementation and datasets are publicly available to support reproducibility and further research.

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