Stochastic Shortest Path problems (SSPs) are traditionally solved by computing each state's cost-to-go by applying Bellman backups. A Bellman backup updates a state's cost-to-go by iterating through every applicable action, computing the cost-to-go after applying each one, and selecting a minimal action's cost-to-go. State-of-the-art algorithms use heuristic functions; these give an initial estimate of costs-to-go, and lets the algorithm apply Bellman backups only to promising states, determined by low estimated costs-to-go. However, each Bellman backup still considers all applicable actions, even if the heuristic tells us that some of these actions are too expensive, with the effect that such algorithms waste time on unhelpful actions. To address this gap we present a technique that uses the heuristic to avoid expensive actions, by reframing heuristic search in terms of linear programming and introducing an efficient implementation of constraint generation for SSPs. We present CG-iLAO*, a new algorithm that adapts iLAO* with our novel technique, and considers only 40% of iLAO*'s actions on many problems, and as few as 1% on some. Consequently, CG-iLAO* computes on average 3.5x fewer costs-to-go for actions than the state-of-the-art iLAO* and LRTDP, enabling it to solve problems faster an average of 2.8x and 3.7x faster, respectively.