# A Convex Optimization Approach to Discrete Optimal Control

## TL;DR

This paper introduces a convex optimization framework for discrete optimal control that incorporates myopic and discrete actions using stochastic subgradients, enabling flexible control policies without altering convex updates.

## Contribution

It extends convex optimization methods to handle discrete and myopic control actions via stochastic and epsilon-subgradients, allowing flexible policy design.

## Key findings

- Decouples control action choice from subgradient selection.
- Provides two classes of discrete control policies, including block-based methods.
- Demonstrates the approach's applicability to systems with action order constraints.

## Abstract

In this paper, we bring the celebrated max-weight features (myopic and discrete actions) to mainstream convex optimization. Myopic actions are important in control because decisions need to be made in an online manner and without knowledge of future events, and discrete actions because many systems have a finite (so non-convex) number of control decisions. For example, whether to transmit a packet or not in communication networks. Our results show that these two features can be encompassed in the subgradient method for the Lagrange dual problem by the use of stochastic and $\epsilon$-subgradients. One of the appealing features of our approach is that it decouples the choice of a control action from a specific choice of subgradient, which allows us to design control policies without changing the underlying convex updates. Two classes of discrete control policies are presented: one that can make discrete actions by looking only at the system's current state, and another that selects actions using blocks. The latter class is useful for handling systems that have constraints on the order in which actions are selected.