This paper 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.

中文翻译：

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无需交叉验证：正则化混合整数问题的分析框架（扩展版）

本文开发了一种在一般混合整数问题 (MIP) 中获得正则化系数最优值的方法。这种方法消除了在现有惩罚技术中执行的交叉验证以获得正则化系数的适当值。我们通过提出一种解决 MIP 的交替方法来实现这一目标。首先，通过正则化，我们将 MIP 转换为更易于数学处理的形式。然后，我们开发了一种迭代算法来更新解决方案以及正则化（惩罚）系数。我们表明我们的更新程序保证了算法的收敛。此外，假设目标函数是连续可微的，我们推导出收敛速度，正则化系数值的下界，以及收敛所需迭代次数的上限。我们在异构网络中使用无线电接入技术 (RAT) 选择问题来衡量我们方法的性能。仿真结果证明了解决方案的接近最优性以及收敛行为与获得的理论界限的一致性。