Enforcing functional inequality constraints such as monotonicity and convexity in neural networks is a fundamental challenge in many industrial and scientific applications. Classical one-sided penalty methods, along with primal-dual methods gated by complementary slackness, provide constraint gradients only at violated locations, resulting in fragile satisfaction. Architectures that guarantee feasibility by construction, on the other hand, remain largely limited to elementary cases and impose additional inductive biases. We introduce neural slack variables, a deep learning native primal-side approach that converts constraint enforcement into a regression problem by coupling the primary network with a jointly learned auxiliary network. The auxiliary network serves as a valid target for the primary network's constraint quantities, inducing feasibility and regularity. Neural slack variables achieve zero measured violations on dense-grid monotonicity and convexity test cases, where penalty and primal-dual baselines leave residual violations, and enable arbitrage-free learning of volatility surfaces, an open industrial challenge in quantitative finance.

翻译：在工业与科学应用中，在神经网络中施加单调性、凸性等函数不等式约束是一项基础性挑战。经典的单侧惩罚方法以及通过互补松弛条件控制的原始-对偶方法仅在违反约束的位置提供梯度信号，导致约束满足的脆弱性。另一方面，通过构造保证可行性的网络架构仍主要局限于基础案例，且会引入额外归纳偏置。我们提出神经松弛变量——一种深度学习原生原始侧方法，通过将主网络与联合学习的辅助网络耦合，将约束强制执行转化为回归问题。辅助网络作为主网络约束量的有效目标值，引导网络满足可行性与正则性。在密集网格单调性与凸性测试案例中，神经松弛变量实现零测量约束违反（而惩罚法与原始-对偶基线方法残留违反值），并能在量化金融领域开放性工业难题——波动率曲面的无套利学习中发挥作用。