Latent-Based Constrained Optimization for Data-Driven Problems

Modern decision-making increasingly relies on data-driven approaches that solve problems by learning patterns from empirical data instead of by using explicit equations or rules. However, integrating these with constrained optimization for optimal decisions remains challenging. Conventional approaches struggle with large-scale, high-dimensional, and unstructured data with implicit relationship with objective and constraints, while deep learning often lacks guarantees for feasibility and optimality. This thesis introduces a latent-based constrained optimization framework to address three critical challenges: (1) identifying active decision variables in learned latent spaces, (2) handling non-convex geometries in optimization, and (3) enforcing hard constraints in deep-learning models without closed-form models.

For the first challenge, we present an entropy-based method to determine which latent factors are genuinely active in a downstream task by jointly learning a Variational Autoencoder (VAE) and a feature selector. This yields a reduced subset of latent dimensions that carry critical information, addressing the importance of isolating decision variables that meaningfully contribute to the objective and constraints in data-driven problems.

For the second challenge, we propose a three-stage uniform transformation module for transforming irregular or Gaussian mixture distributions within the latent space into a more tractable representation that achieves a nearly-convex set. By mitigating issues like over-pruning, posterior collapse, and misaligned distributions, this approach better preserves essential data features and improves downstream optimization performance.

Finally, for the last challenge, we analyzed and modified a constraint-priority filter method that operates as a bi-objective problem in the latent domain to impose hard constraints directly on a deep surrogate model. Unlike penalty-based methods, this approach systematically balances objective improvement against constraint satisfaction, offering strict feasibility guarantees for problems where constraints may be only implicitly defined via data.

Comprehensive experiments, including synthetic and real datasets, well-recognized benchmarks, different deep network architectures, and real-world scenarios, have validated the effectiveness of our proposed framework. Results showed improved reconstruction quality, reliable disentanglement or feature extraction, and consistent satisfaction of hard constraints in inverse problems.

Overall, this thesis establishes a general framework that integrates latent representation learning and constrained optimization, offering both theoretical insights — such as the link between entropy regularization and disentanglement — and practical tools for automating decision-making in data-rich, constraint-heavy domains. By harmonizing the representational power of deep learning with the rigor of classical optimization, it paves the way for safer, more reliable artificial intelligence systems in science, engineering, and beyond.