Riemannian Constrained Optimization

RCO gains efficiency over evolutionary methods because it uses first-order information. RCO combines manifold operations with Gumbel straight-through estimation and budget-constrained dynamic programming. RCO uses dynamic programming in the forward pass and a straight-through estimator in the backward pass. The algorithm keeps discrete candidates budget-feasible while constraining the continuous logit distribution through projection and retraction. The final assignment is computed by solving the same DP using log probabilities without noise or temperature.