Contextual Strongly Convex Simulation Optimization: Optimize then Predict with Inexact Solutions
Nifei Lin, Heng Luo, L. Jeff Hong
TL;DR
This work develops an optimize-then-predict framework for contextual strongly convex simulation optimization (CSCSO) that accommodates inexact offline solutions. By modeling the optimal-solution function θ*(x) as smooth, the authors design an offline-online pipeline: solve SO problems at a quasi-uniform covariate design with budget Γ, then apply smoothing to predict the optimal decision for new covariates. They quantify the impact of solution bias and variance arising from finite offline budgets using PR-SGD and analyze four smoothing techniques (kNN, KS, LR, KRR) to derive optimal sample-allocation rules and convergence rates, showing that under appropriate settings the MSE and the optimality gap can achieve rates like Γ^{−2/(d+2)} or Γ^{−1}, depending on the method and problem smoothness. Numerical results on a newsvendor example demonstrate that the proposed allocation rules and KRR/LR smoothing achieve strong predictive performance and robustness to high dimensional covariates, illustrating the practical value of OTP with inexact solutions for real-time contextual decisions.
Abstract
In this work, we study contextual strongly convex simulation optimization and adopt an "optimize then predict" (OTP) approach for real-time decision making. In the offline stage, simulation optimization is conducted across a set of covariates to approximate the optimal-solution function; in the online stage, decisions are obtained by evaluating this approximation at the observed covariate. The central theoretical challenge is to understand how the inexactness of solutions generated by simulation-optimization algorithms affects the optimality gap, which is overlooked in existing studies. To address this, we develop a unified analysis framework that explicitly accounts for both solution bias and variance. Using Polyak-Ruppert averaging SGD as an illustrative simulation-optimization algorithm, we analyze the optimality gap of OTP under four representative smoothing techniques: $k$ nearest neighbor, kernel smoothing, linear regression, and kernel ridge regression. We establish convergence rates, derive the optimal allocation of the computational budget $Γ$ between the number of design covariates and the per-covariate simulation effort, and demonstrate the convergence rate can approximately achieve $Γ^{-1}$ under appropriate smoothing technique and sample-allocation rule. Finally, through a numerical study, we validate the theoretical findings and demonstrate the effectiveness and practical value of the proposed approach.