# Simulation-Based Optimization over Discrete Spaces Using Projection to Continuous Latent Spaces

## TL;DR

This paper introduces a method to optimize complex systems with discrete choices by transforming them into continuous spaces for efficient simulation-based optimization.

## Contribution

A novel approach using Variational AutoEncoders to project discrete decision spaces into continuous latent spaces for Bayesian optimization.

## Key findings

- Transforming discrete spaces into continuous latent spaces enables efficient Bayesian optimization.
- The method was successfully applied to design complex distillation systems with promising results.

## Abstract

Simulation-based optimization of complex systems over
discrete
decision spaces is a challenging computational problem. Specifically,
discrete decision spaces lead to a combinatorial explosion of possible
alternatives, making it computationally prohibitive to perform simulations
for all possible combinations. In this work, we present a new approach
to handle these issues by transforming/projecting the discrete decision
space into a continuous latent space using a probabilistic model known
as Variational AutoEncoders. The transformation of the decision space
facilitates the implementation of Bayesian optimization (BO), which
is an efficient approach that strategically navigates the space to
reduce the number of expensive simulations. Here, the key observation
is that points in the latent space correspond to decisions in the
original mixed-discrete space, but the latent space is much easier
to navigate using BO. We illustrate the benefits of our approach through
a couple of case studies that aim to design complex distillation systems:
the recovery of caprylic acid from water by liquid–liquid extraction
and the separation of an azeotropic mixture using a thermally couple
column known as an extractive dividing wall column.