# Accelerated Point-wise Maximum Approach to Approximate Dynamic Programming

## TL;DR

This paper introduces a novel approximate dynamic programming method that iteratively constructs lower bounds on the value function using a point-wise maximum approach, employing a gradient ascent algorithm with convergence guarantees.

## Contribution

It proposes a new approach to approximate dynamic programming that maximizes point-wise maximums of value functions, with a gradient ascent method and convergence analysis.

## Key findings

- Computes tighter sub-optimality bounds than existing methods.
- Requires less computation time for the same or better bounds.
- Provides convergence guarantees for the proposed algorithm.

## Abstract

We describe an approximate dynamic programming approach to compute lower bounds on the optimal value function for a discrete time, continuous space, infinite horizon setting. The approach iteratively constructs a family of lower bounding approximate value functions by using the so-called Bellman inequality. The novelty of our approach is that, at each iteration, we aim to compute an approximate value function that maximizes the point-wise maximum taken with the family of approximate value functions computed thus far. This leads to a non-convex objective, and we propose a gradient ascent algorithm to find stationary points by solving a sequence of convex optimization problems. We provide convergence guarantees for our algorithm and an interpretation for how the gradient computation relates to the state relevance weighting parameter appearing in related approximate dynamic programming approaches. We demonstrate through numerical examples that, when compared to existing approaches, the algorithm we propose computes tighter sub-optimality bounds with less computation time.