Ideal formulations for constrained convex optimization problems with indicator variables
This work addresses the problem of efficiently solving constrained convex optimization with indicator variables for applications like modern regression, offering incremental improvements in convexification methods.
The paper tackles the convexification of optimization problems with indicator variables and combinatorial constraints, deriving ideal convex hull descriptions for various constraint types and demonstrating improved relaxation quality in regression experiments on real datasets.
Motivated by modern regression applications, in this paper, we study the convexification of a class of convex optimization problems with indicator variables and combinatorial constraints on the indicators. Unlike most of the previous work on convexification of sparse regression problems, we simultaneously consider the nonlinear non-separable objective, indicator variables, and combinatorial constraints. Specifically, we give the convex hull description of the epigraph of the composition of a one-dimensional convex function and an affine function under arbitrary combinatorial constraints. As special cases of this result, we derive ideal convexifications for problems with hierarchy, multi-collinearity, and sparsity constraints. Moreover, we also give a short proof that for a separable objective function, the perspective reformulation is ideal independent from the constraints of the problem. Our computational experiments with regression problems under hierarchy constraints on real datasets demonstrate the potential of the proposed approach in improving the relaxation quality without significant computational overhead.