The Asymmetric Traveling Salesman Problem (ATSP) ranks among the most fundamental and notoriously difficult problems in combinatorial optimization. We propose a novel continuous relaxation framework for the Asymmetric Traveling Salesman Problem (ATSP) by leveraging differentiable constraints that encourage acyclic structures and valid permutations. Our approach integrates a differentiable trace-based Directed Acyclic Graph (DAG) constraint with a doubly stochastic matrix relaxation of the assignment problem, enabling gradient-based optimization over soft permutations. We develop a projected exponentiated gradient method with adaptive step size to minimize tour cost while satisfying the relaxed constraints. To recover high-quality discrete tours, we introduce a greedy post-processing procedure that iteratively corrects subtours using cost-aware cycle merging. Our method achieves state-of-the-art performance on standard asymmetric TSP benchmarks and demonstrates competitive scalability and accuracy, particularly on large or asymmetric instances where heuristic solvers such as LKH-3 struggle.