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Many practical problems are too difficult to solve optimally, motivating the need to found suboptimal solutions, particularly those with bounds on the final solution quality. Algorithms like Weighted A*, A*-epsilon, Optimistic Search, EES, and DPS have been developed to find suboptimal solutions with solution quality that is within a constant bound of the optimal solution. However, with the exception of weighted A*, all of these algorithms require performing node re-expansions during search. This paper explores the properties of priority functions that can find bounded suboptimal solution without requiring node re-expansions. After general bounds are developed, two new convex priority functions are developed that can outperform weighted A*.