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Penalized regression methods, such as lasso and elastic net, are used in many biomedical applications when simultaneous regression coefficient estimation and variable selection is desired. However, missing data complicates the implementation of these methods, particularly when missingness is handled using multiple imputation. Applying a variable selection algorithm on each imputed dataset will likely lead to different sets of selected predictors, making it difficult to ascertain a final active set without resorting to ad hoc combination rules. In this paper we consider a general class of penalized objective functions which, by construction, force selection of the same variables across multiply-imputed datasets. By pooling objective functions across imputations, optimization is then performed jointly over all imputed datasets rather than separately for each dataset. We consider two objective function formulations that exist in the literature, which we will refer to as "stacked" and "grouped" objective functions. Building on existing work, we (a) derive and implement efficient cyclic coordinate descent and majorization-minimization optimization algorithms for both continuous and binary outcome data, (b) incorporate adaptive shrinkage penalties, (c) compare these methods through simulation, and (d) develop an R package miselect for easy implementation. Simulations demonstrate that the "stacked" objective function approaches tend to be more computationally efficient and have better estimation and selection properties. We apply these methods to data from the University of Michigan ALS Patients Repository (UMAPR) which aims to identify the association between persistent organic pollutants and ALS risk.