  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.
