# COIL: Constrained Optimization in Learned Latent Space: Learning Representations for Valid Solutions

arXiv: https://arxiv.org/abs/2202.02163 · HTML: https://mlsanity.com/paper/2202.02163/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 40 references that are in the map.
- First posted on arXiv 4 Feb 2022.

## Builds on (most cited first, 9 of 9)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Illuminating search spaces by mapping elites](https://mlsanity.com/paper/1504.04909/) (2015)
- [Quality and Diversity Optimization: A Unifying Modular Framework](https://mlsanity.com/paper/1708.09251/) (2017)
- [Evolving Mario Levels in the Latent Space of a Deep Convolutional Generative Adversarial Network](https://mlsanity.com/paper/1805.00728/) (2018)
- [Differentiable Quality Diversity](https://mlsanity.com/paper/2106.03894/) (2021)
- [Policy Manifold Search: Exploring the Manifold Hypothesis for Diversity-based Neuroevolution](https://mlsanity.com/paper/2104.13424/) (2021)
- [Benchmarking Evolutionary Algorithms For Single Objective Real-valued Constrained Optimization - A Critical Review](https://mlsanity.com/paper/1806.04563/) (2018)
- [Expressivity of Parameterized and Data-driven Representations in Quality Diversity Search](https://mlsanity.com/paper/2105.04247/) (2021)

## Built on by (most cited first, 10 of 10)

- [SCAPE: Searching Conceptual Architecture Prompts using Evolution](https://mlsanity.com/paper/2402.00089/) (2024)
- [Using a Variational Autoencoder to Learn Valid Search Spaces of Safely Monitored Autonomous Robots for Last-Mile Delivery](https://mlsanity.com/paper/2303.03211/) (2023)
- [Full Domain Analysis in Fluid Dynamics](https://mlsanity.com/paper/2505.22275/) (2025)
- [Efficient Quality Diversity Optimization of 3D Buildings through 2D Pre-optimization](https://mlsanity.com/paper/2303.15896/) (2023)
- [Inverse Optimization Latent Variable Models for Learning Costs Applied to Route Problems](https://mlsanity.com/paper/2509.15999/) (2025)
- [Binary Latent Protein Fitness Landscapes for Quantum Annealing Optimization](https://mlsanity.com/paper/2603.17247/) (2026)
- [Q-BIOLAT: Binary Latent Protein Fitness Landscapes for QUBO-Based Optimization](https://mlsanity.com/paper/2603.27526/) (2026)
- [LEGO: Latent-space Exploration for Geometry-aware Optimization of Humanoid Kinematic Design](https://mlsanity.com/paper/2604.08636/) (2026)
- [A Multi-stage Constrained Optimization Framework for Data-driven Problems](https://mlsanity.com/paper/2607.23480/) (2026)
- [Soft-Constrained Optimization of Latent Space in Variational Autoencoders](https://mlsanity.com/paper/2607.23751/) (2026)

## Abstract (arXiv)

Constrained optimization problems can be difficult because their search spaces have properties not conducive to search, e.g., multimodality, discontinuities, or deception. To address such difficulties, considerable research has been performed on creating novel evolutionary algorithms or specialized genetic operators. However, if the representation that defined the search space could be altered such that it only permitted valid solutions that satisfied the constraints, the task of finding the optimal would be made more feasible without any need for specialized optimization algorithms. We propose Constrained Optimization in Latent Space (COIL), which uses a VAE to generate a learned latent representation from a dataset comprising samples from the valid region of the search space according to a constraint, thus enabling the optimizer to find the objective in the new space defined by the learned representation. Preliminary experiments show promise: compared to an identical GA using a standard representation that cannot meet the constraints or find fit solutions, COIL with its learned latent representation can perfectly satisfy different types of constraints while finding high-fitness solutions.