Certified Primal-Dual State Reuse for Sequential Robust Tensor Sampling

Tensor sampling often arises as a sequence of related tasks in which sensing factors and uncertainty change while earlier optimization states remain informative. Existing samplers typically solve each task independently, whereas heuristic warm starts provide no test of whether a stored state remains valid after the task changes. We propose Variation-Aware Primal-Dual Memory Robust Frank-Wolfe (VPM-RFW), a projection-free algorithm that stores outer sampling designs and inner primal-dual states and validates their reuse through transfer bounds across tasks. These bounds control changes in the robust value, KKT state, and Frank-Wolfe gap. They allow the method to return a stored point that meets the current stationarity tolerance, continue from a reusable state, or fall back to a cold start. Under verified regularity and provided that no fallback occurs, the continuous output satisfies FW stationarity at tolerance $ϵ$ with a weighted solver oracle bound. Feasible binary rounding has a separate bound on the objective value. On the tested synthetic sequences and KSC data, reusing states required less measured solver work than cold starts.