Constrained Optimization with Decision-Dependent Distributions

TL;DR

This paper studies stochastic optimization problems where both the data distributions and constraints depend on decisions, introducing new conditions for equilibrium existence and proposing algorithms with convergence guarantees, validated through real-world experiments.

Contribution

It extends decision-dependent distribution analysis to include dynamic constraints, providing theoretical conditions and algorithms for convergence in this more general setting.

Findings

Established a sufficient condition for constrained equilibrium existence.

Proposed and analyzed two algorithms with convergence guarantees.

Validated methods through real-world market and pricing experiments.

Abstract

In this paper we deal with stochastic optimization problems where the data distributions change in response to the decision variables. Traditionally, the study of optimization problems with decision-dependent distributions has assumed either the absence of constraints or fixed constraints. This work considers a more general setting where the constraints can also dynamically adjust in response to changes in the decision variables. Specifically, we consider linear constraints and analyze the effect of decision-dependent distributions in both the objective function and constraints. Firstly, we establish a sufficient condition for the existence of a constrained equilibrium point, at which the distributions remain invariant under retraining. Morevoer, we propose and analyze two algorithms: repeated constrained optimization and repeated dual ascent. For each algorithm, we provide sufficient conditions for convergence to the constrained equilibrium point. Furthermore, we explore the relationship between the equilibrium point and the optimal point for the constrained decision-dependent optimization problem. Notably, our results encompass previous findings as special cases when the constraints remain fixed. To show the effectiveness of our theoretical analysis, we provide numerical experiments on both a market problem and a dynamic pricing problem for parking based on real-world data.