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Application-specific, parameterized local search algorithms (PLSAs), in which optimization accuracy can be traded off with run time, arise naturally in many optimization contexts. We introduce a novel approach, called simulated heating, for systematically integrating parameterized local search into evolutionary algorithms (EAs). Using the framework of simulated heating, we investigate both static and dynamic strategies for systematically managing the tradeoff between PLSA accuracy and optimization effort. Our goal is to achieve maximum solution quality within a fixed optimization time budget. We show that the simulated heating technique better utilizes the given optimization time resources than standard hybrid methods that employ fixed parameters, and that the technique is less sensitive to these parameter settings. We apply this framework to three different optimization problems, compare our results to the standard hybrid methods, and show quantitatively that careful management of this tradeoff is necessary to achieve the full potential of an EA/PLSA combination.