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The design of optimal multi-resolution scalar quantizers using the generalized Lloyd method was proposed by Brunk and Farvardin for the case of squared error distortion. Since the algorithm details heavily rely on the quadratic expression of the error function, its extension to general error functions faces some challenges, especially at the encoder optimization step. In this work we show how these challenges can be overcome for any convex difference distortion measure, under the assumption that all quantizer cells are convex (i.e., intervals), and present an efficient algorithm for optimal encoder partition computation. The proposed algorithm is faster than the algorithm used by Brunk and Farvardin. Moreover, it can also be applied to channel-optimized and to multiple description scalar quantizer design with squared error distortion, and it outperforms in speed the previous encoder optimization algorithms proposed for these problems.