The compatibility constant plays an important role in evaluating the prediction error of the lasso in high-dimensional settings. However, the computation of the compatibility constant is generally difficult because it is a complicated nonconvex optimization problem. In this study, we present a numerical approach to compute the compatibility constant when the support of true regression coefficients is given. We show that the optimization problem reduces to a quadratic programming (QP) once the signs of the nonzero coefficients are specified. In this case, the compatibility constant can be obtained by solving QPs for all possible sign combinations. We also formulate a mixed-integer QP (MIQP) approach that can be applied when the number of true nonzero coefficients is large. We investigate the finite-sample behavior of the compatibility constant for simulated data under various parameter settings and compare the prediction error with its theoretical upper bound. The behavior of the compatibility constant in finite samples is also investigated through a real data analysis.

翻译：兼容常数在高维设定下评估lasso预测误差中扮演重要角色。然而，兼容常数的计算通常很困难，因为这是一个复杂的非凸优化问题。在本研究中，我们提出了一种数值方法，用于在已知真实回归系数支撑集的情况下计算兼容常数。我们证明，一旦非零系数的符号被指定，优化问题可简化为二次规划（QP）。在这种情况下，可通过求解所有可能符号组合的QP来获得兼容常数。我们还提出了一种混合整数QP（MIQP）方法，适用于真实非零系数数量较大的情形。我们研究了在不同参数设定下模拟数据中兼容常数的有限样本行为，并将其预测误差与理论上限进行了比较。通过实际数据分析，我们进一步探究了兼容常数在有限样本中的表现。