Neural Slack Variables Enforce Strict Shape Constraints in Networks

This research introduces a novel deep learning technique called "neural slack variables" designed to robustly enforce functional inequality constraints within neural networks. Constraints such as monotonicity and convexity are fundamental in many scientific and industrial applications, but traditional methods like one-sided penalties often result in residual violations. The proposed approach transforms constraint enforcement into a regression problem. It couples the primary neural network with a jointly learned auxiliary network, which serves as a valid target for the primary network's constraint quantities. This mechanism effectively induces feasibility and regularity in the network's output. Empirical evaluations demonstrate that neural slack variables achieve zero measured violations on dense-grid test cases for monotonicity and convexity, a significant improvement over penalty and primal-dual baselines that typically leave residual violations. A key application highlighted is the ability to enable arbitrage-free learning of volatility surfaces, addressing a long-standing challenge in quantitative finance.