Online Mixed Discrete and Continuous Optimization: Algorithms, Regret Analysis and Applications

TL;DR

This paper introduces algorithms for online mixed discrete and continuous optimization, providing regret guarantees and demonstrating their effectiveness through theoretical analysis and numerical experiments across various practical applications.

Contribution

The paper proposes novel algorithms for online mixed discrete and continuous optimization with proven sublinear regret bounds, extending the applicability to diverse real-world problems.

Findings

Algorithms achieve sublinear regret in T

Validated effectiveness through numerical experiments

Applicable to a wide range of practical problems

Abstract

We study an online mixed discrete and continuous optimization problem where a decision maker interacts with an unknown environment for a number of $T$ rounds. At each round, the decision maker needs to first jointly choose a discrete and a continuous actions and then receives a reward associated with the chosen actions. The goal for the decision maker is to maximize the accumulative reward after $T$ rounds. We propose algorithms to solve the online mixed discrete and continuous optimization problem and prove that the algorithms yield sublinear regret in $T$. We show that a wide range of applications in practice fit into the framework of the online mixed discrete and continuous optimization problem, and apply the proposed algorithms to solve these applications with regret guarantees. We validate our theoretical results with numerical experiments.