Many real-world optimization problems involve uncertain parameters with probability distributions that can be estimated using contextual feature information. In contrast to the standard approach of first estimating the distribution of uncertain parameters and then optimizing the objective based on the estimation, we propose an integrated conditional estimation-optimization (ICEO) framework that estimates the underlying conditional distribution of the random parameter while considering the structure of the optimization problem. We directly model the relationship between the conditional distribution of the random parameter and the contextual features, and then estimate the probabilistic model with an objective that aligns with the downstream optimization problem. We show that our ICEO approach is asymptotically consistent under moderate regularity conditions and further provide finite performance guarantees in the form of generalization bounds. Computationally, performing estimation with the ICEO approach is a non-convex and often non-differentiable optimization problem. We propose a general methodology for approximating the potentially non-differentiable mapping from estimated conditional distribution to the optimal decision by a differentiable function, which greatly improves the performance of gradient-based algorithms applied to the non-convex problem. We also provide a polynomial optimization solution approach in the semi-algebraic case. Numerical experiments are also conducted to show the empirical success of our approach in different situations including with limited data samples and model mismatches.

翻译：许多实际优化问题涉及具有概率分布的不确定参数，这些分布可利用情境特征信息进行估计。与传统先估计不确定参数分布、再基于估计结果优化目标的方法不同，我们提出了一个集成条件估计-优化（ICEO）框架，该框架在考虑优化问题结构的同时，对随机参数的条件分布进行估计。我们直接建模随机参数条件分布与情境特征之间的关系，并以与下游优化目标对齐的方式估计概率模型。我们证明了在适度正则条件下，ICEO方法具有渐近一致性，并进一步以泛化界的形式提供了有限样本性能保障。在计算层面，采用ICEO方法进行估计是一个非凸且通常不可微的优化问题。我们提出了一种通用方法，将可能不可微的映射（从估计的条件分布到最优决策）通过可微函数近似，从而显著提升梯度算法应用于非凸问题的性能。在半代数情形下，我们还提供了一种多项式优化求解方案。通过数值实验，我们展示了该方法在包括有限数据样本和模型失配等不同情境下的实证效果。