We propose an inference procedure for a class of estimators defined as the solutions to linear and convex quadratic programming problems in which the coefficients in both the objective function and the constraints of the problem are estimated from data and hence involve sampling error. We argue that the Karush–Kuhn–Tucker conditions that characterize the solutions to these programming problems can be treated as moment conditions; by doing so, we transform the problem of inference on the solution to a constrained optimization problem (which is non-standard) into one involving inference on inequalities with pre-estimated coefficients, which is better understood. Our approach is valid regardless of whether the problem has a unique solution or multiple solutions. We apply our method to various portfolio selection models, in which the confidence sets can be non-convex, lower-dimensional manifolds.

我们提出了一类估计器的推理过程，定义为线性和凸二次规划问题的解决方案，其中目标函数中的系数和问题的约束都是从数据中估计出来的，因此涉及采样误差。我们认为，表征这些规划问题的解决方案的 Karush-Kuhn-Tucker 条件可以被视为矩条件；通过这样做，我们将约束优化问题（这是非标准的）解的推理问题转换为一个涉及具有预估计系数的不等式推理的问题，这更好理解。无论问题有唯一解决方案还是多个解决方案，我们的方法都是有效的。我们将我们的方法应用于各种投资组合选择模型，