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

TL;DR

This paper develops a theoretical framework for analyzing the impact of inexact solutions in a simulation optimization approach for real-time decision making, providing convergence rates and practical insights.

Contribution

It introduces a unified analysis framework for inexact solutions in contextual strongly convex simulation optimization, including convergence rates and optimal resource allocation strategies.

Findings

Convergence rate can approach rac{ ext{Gamma}}{1} under certain conditions.

Optimal allocation of computational budget balances covariate and simulation efforts.

Numerical results validate theoretical convergence and practical effectiveness.

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 $\Gamma$ 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.