  Performative prediction is a recently proposed framework where predictions
guide decision-making and hence influence future data distributions. Such
performative phenomena are ubiquitous in various areas, such as transportation,
finance, public policy, and recommendation systems. To date, work on
performative prediction has only focused on unconstrained scenarios, neglecting
the fact that many real-world learning problems are subject to constraints.
This paper bridges this gap by studying performative prediction under
inequality constraints. Unlike most existing work that provides only
performative stable points, we aim to find the optimal solutions. Anticipating
performative gradients is a challenging task, due to the agnostic performative
effect on data distributions. To address this issue, we first develop a robust
primal-dual framework that requires only approximate gradients up to a certain
accuracy, yet delivers the same order of performance as the stochastic
primal-dual algorithm without performativity. Based on this framework, we then
propose an adaptive primal-dual algorithm for location families. Our analysis
demonstrates that the proposed adaptive primal-dual algorithm attains
$\ca{O}(\sqrt{T})$ regret and constraint violations, using only $\sqrt{T} + 2T$
samples, where $T$ is the time horizon. To our best knowledge, this is the
first study and analysis on the optimality of the performative prediction
problem under inequality constraints. Finally, we validate the effectiveness of
our algorithm and theoretical results through numerical simulations.
