Amplitude Modulated Riemannian Optimization for QAP

We introduce a novel approach for solving the Quadratic Assignment Problem (QAP) by combining Riemannian optimization and control of soft-spin amplitude. By reformulating the QAP as an optimization problem on the Stiefel manifold, we leverage its geometric structure to define Riemannian gradients over continuous variables while inherently satisfying orthogonality constraints. Additional permutation matrix constraints are enforced using auxiliary variables within a descent-ascent framework, ensuring that solutions remain within the feasible set. Numerical simulations demonstrate the effectiveness of our method in finding globally optimal solutions.