Recent advances in genomics have underscored the surprising ubiquity of DNA copy number variation (CNV). Fortunately, modern genotyping platforms also detect CNVs with fairly high reliability. Hidden Markov models and algorithms have played a dominant role in the interpretation of CNV data. Here we explore CNV reconstruction via estimation with a fused-lasso penalty as suggested by Tibshirani and Wang [Biostatistics 9 (2008) 18-29]. We mount a fresh attack on this difficult optimization problem by the following: (a) changing the penalty terms slightly by substituting a smooth approximation to the absolute value function, (b) designing and implementing a new MM (majorization-minimization) algorithm, and (c) applying a fast version of Newton's method to jointly update all model parameters. Together these changes enable us to minimize the fused-lasso criterion in a highly effective way.We also reframe the reconstruction problem in terms of imputation via discrete optimization. This approach is easier and more accurate than parameter estimation because it relies on the fact that only a handful of possible copy number states exist at each SNP. The dynamic programming framework has the added bonus of exploiting information that the current fused-lasso approach ignores. The accuracy of our imputations is comparable to that of hidden Markov models at a substantially lower computational cost.