The connection between regularization and min–max robustification in the presence of unobservable covariate measurement errors in linear mixed models is addressed. We prove that regularized model parameter estimation is equivalent to robust loss minimization under a min–max approach. On the example of the LASSO, Ridge regression, and the Elastic Net, we derive uncertainty sets that characterize the feasible noise that can be added to a given estimation problem. These sets allow us to determine measurement error bounds without distribution assumptions. A conservative Jackknife estimator of the mean squared error in this setting is proposed. We further derive conditions under which min-max robust estimation of model parameters is consistent. The theoretical findings are supported by a Monte Carlo simulation study under multiple measurement error scenarios.
Cardinality-constrained optimization problems are notoriously hard to solve in both theory and practice. However, as famous examples, such as the sparse portfolio optimization and best subset selection problems, show, this class is extremely important in real-world applications. In this paper, we apply a penalty alternating direction method to these problems. The key idea is to split the problem along its discrete-continuous structure to obtain two subproblems that are much easier to solve than the original problem. In addition, the coupling between these subproblems is achieved via a classic penalty framework. The method can be seen as a primal heuristic for which convergence results are readily available from the literature. In our extensive computational study, we first show that the method is competitive to a commercial mixed-integer program solver for the portfolio optimization problem. On these instances, we also test a variant of our approach that uses a perspective reformulation of the problem. Regarding the best subset selection problem, it turns out that our method significantly outperforms commercial solvers and it is at least competitive to state-of-the-art methods from the literature.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.