# Personalized Optimization with User's Feedback

## TL;DR

This paper introduces an online algorithm that balances known time-varying costs and unknown user satisfaction, using Gaussian processes to learn and adapt in real-time, with applications demonstrated in vehicle platooning.

## Contribution

It develops a novel online optimization method that simultaneously learns user satisfaction and tracks optimal solutions in dynamic environments.

## Key findings

- The algorithm effectively balances performance and user satisfaction.
- It achieves no-regret learning of the user's utility function.
- Numerical examples demonstrate practical applicability in vehicle platooning.

## Abstract

This paper develops an online algorithm to solve a time-varying optimization problem with an objective that comprises a known time-varying cost and an unknown function. This problem structure arises in a number of engineering systems and cyber-physical systems where the known function captures time-varying engineering costs, and the unknown function models user's satisfaction; in this context, the objective is to strike a balance between given performance metrics and user's satisfaction. Key challenges related to the problem at hand are related to (1) the time variability of the problem, and (2) the fact that learning of the user's utility function is performed concurrently with the execution of the online algorithm. This paper leverages Gaussian processes (GP) to learn the unknown cost function from noisy functional evaluation and build pertinent upper confidence bounds. Using the GP formalism, the paper then advocates time-varying optimization tools to design an online algorithm that exhibits tracking of the oracle-based optimal trajectory within an error ball, while learning the user's satisfaction function with no-regret. The algorithmic steps are inexact, to account for possible limited computational budgets or real-time implementation considerations. Numerical examples are illustrated based on a problem related to vehicle platooning.