Online Projected Gradient Descent for Stochastic Optimization with\n Decision-Dependent Distributions

This paper investigates the problem of tracking solutions of stochastic\noptimization problems with time-varying costs that depend on random variables\nwith decision-dependent distributions. In this context, we propose the use of\nan online stochastic gradient descent method to solve the optimization, and we\nprovide explicit bounds in expectation and in high probability for the distance\nbetween the optimizers and the points generated by the algorithm. In\nparticular, we show that when the gradient error due to sampling is modeled as\na sub-Weibull random variable, then the tracking error is ultimately bounded in\nexpectation and in high probability. The theoretical findings are validated via\nnumerical simulations in the context of charging optimization of a fleet of\nelectric vehicles.\n