Projected Gradient Descent for Constrained Decision-Dependent Optimization

TL;DR

This paper introduces RPGD, a new iterative method for decision-dependent optimization that maintains feasibility and converges under certain conditions, demonstrated through market and pricing experiments.

Contribution

The paper proposes RPGD, a novel projected gradient method for decision-dependent problems, with convergence analysis and advantages over dual ascent methods.

Findings

RPGD maintains feasibility during optimization.

RPGD converges to the constrained equilibrium under certain conditions.

Numerical experiments validate the effectiveness of RPGD.

Abstract

This paper considers the decision-dependent optimization problem, where the data distributions react in response to decisions affecting both the objective function and linear constraints. We propose a new method termed repeated projected gradient descent (RPGD), which iteratively projects points onto evolving feasible sets throughout the optimization process. To analyze the impact of varying projection sets, we show a Lipschitz continuity property of projections onto varying sets with an explicitly given Lipschitz constant. Leveraging this property, we provide sufficient conditions for the convergence of RPGD to the constrained equilibrium point. Compared to the existing dual ascent method, RPGD ensures continuous feasibility throughout the optimization process and reduces the computational burden. We validate our results through numerical experiments on a market problem and dynamic pricing problem.