  We consider the problem of parameter estimation in slowly varying regression
models with sparsity constraints. We formulate the problem as a mixed integer
optimization problem and demonstrate that it can be reformulated exactly as a
binary convex optimization problem through a novel exact relaxation. The
relaxation utilizes a new equality on Moore-Penrose inverses that convexifies
the non-convex objective function while coinciding with the original objective
on all feasible binary points. This allows us to solve the problem
significantly more efficiently and to provable optimality using a cutting
plane-type algorithm. We develop a highly optimized implementation of such
algorithm, which substantially improves upon the asymptotic computational
complexity of a straightforward implementation. We further develop a heuristic
method that is guaranteed to produce a feasible solution and, as we empirically
illustrate, generates high quality warm-start solutions for the binary
optimization problem. We show, on both synthetic and real-world datasets, that
the resulting algorithm outperforms competing formulations in comparable times
across a variety of metrics including out-of-sample predictive performance,
support recovery accuracy, and false positive rate. The algorithm enables us to
train models with 10,000s of parameters, is robust to noise, and able to
effectively capture the underlying slowly changing support of the data
generating process.
