Temporal‑difference (TD) critics normally regress onto a bootstrapped target and rely on a target network. Lifted Bellman Linear Programming (LBLP) instead formulates the in‑sample optimal values of offline RL as the solution of a linear program over the dataset. Its neural solver ALBUM relaxes the constraints with hinge penalties using fixed coefficients, and its guarantees depend on those pre‑chosen coefficients. We solve the same program with the method of multipliers: the coefficients become multipliers learned by dual ascent, with a single penalty parameter $\rho$. In the convex case the fixed points are exactly the KKT pairs of LBLP, and at every stationary point the applied constraint weights balance the objective precisely. The critic can be paired with a Gaussian actor, a flow actor, or best‑of‑N sampling; for the Gaussian actor we propose an advantage‑normalized DDPG+BC objective. Using a single $Q$ and $V$ and no target network, AL‑LBLP with best‑of‑N achieves 60.1% average success on the 50 state‑based OGBench tasks. On long‑horizon benchmarks it reaches 81.5% on humanoidmaze‑giant and 79.8% on antmaze‑giant, and it remains stable up to batch size 8192.\n\nReview: By integrating an augmented Lagrangian scheme into offline RL’s linear programming formulation, the paper eliminates the need for target networks and enables adaptive constraint weighting via dual learning, demonstrating strong robustness and scalability on large offline datasets.