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We present a rigorous framework that defines a class of net weighting schemes in which unconstrained minimization is successively performed on a weighted objective. We show that, provided certain goals are met in the unconstrained minimization, these net weighting schemes are guaranteed to converge to the optimal solution of the original timing-constrained placement problem. These are the first results that provide conditions under which a net weighting scheme will converge to a timing optimal placement. We then identify several weighting schemes that satisfy the given convergence properties and implement them, with promising results: a modification of the weighting scheme given in results in consistently improved delay over the original, 4% on average, without increase in computation time.