No Cross-Validation Required: An Analytical Framework for Regularized Mixed-Integer Problems

This letter develops a method to obtain the optimal value for the regularization coefficient in a general mixed-integer problem (MIP). This approach eliminates the cross-validation performed in the existing penalty techniques to obtain a proper value for the regularization coefficient. We obtain this goal by proposing an alternating method to solve MIPs. First, via regularization, we convert the MIP into a more mathematically tractable form. Then, we develop an iterative algorithm to update the solution along with the regularization (penalty) coefficient. We show that our update procedure guarantees the convergence of the algorithm. Moreover, assuming the objective function is continuously differentiable, we derive the convergence rate, a lower bound on the value of regularization coefficient, and an upper bound on the number of iterations required for the convergence. We use a radio access technology (RAT) selection problem in a heterogeneous network to benchmark the performance of our method. Simulation results demonstrate near-optimality of the solution and consistency of the convergence behavior with obtained theoretical bounds.