  Variable selection is an old and pervasive problem in regression analysis.
One solution is to impose a lasso penalty to shrink parameter estimates toward
zero and perform continuous model selection. The lasso-penalized mixture of
linear regressions model (L-MLR) is a class of regularization methods for the
model selection problem in the fixed number of variables setting. In this
article, we propose a new algorithm for the maximum penalized-likelihood
estimation of the L-MLR model. This algorithm is constructed via the
minorization--maximization algorithm paradigm. Such a construction allows for
coordinate-wise updates of the parameter components, and produces globally
convergent sequences of estimates that generate monotonic sequences of
penalized log-likelihood values. These three features are missing in the
previously presented approximate expectation-maximization algorithms. The
previous difficulty in producing a globally convergent algorithm for the
maximum penalized-likelihood estimation of the L-MLR model is due to the
intractability of finding exact updates for the mixture model mixing
proportions in the maximization-step. In our algorithm, we solve this issue by
showing that it can be converted into a polynomial root finding problem. Our
solution to this problem involves a polynomial basis conversion that is
interesting in its own right. The method is tested in simulation and with an
application to Major League Baseball salary data from the 1990s and the present
day. We explore the concept of whether player salaries are associated with
batting performance.
