Contextual Strongly Convex Simulation Optimization: Optimize then Predict with Inexact Solutions

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 Gamma between the number of design covariates and the per-covariate simulation effort, and demonstrate the convergence rate can approximately achieve Gamma-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.