Abstract

A new computationally simple method of imposing hard convex constraints on the neural network output values is proposed. The key idea is to map a latent vector to a point that is guaranteed to be inside the feasible set defined by a set of constraints. The mapping is implemented by the additional neural network layer with constraints for output. The method proposed is extended to the case when constraints are imposed not only on the output vectors, but also on joint constraints depending on inputs. The projection approach to imposing constraints on outputs can simply be implemented in the framework of the proposed method. It is shown how to incorporate different types of constraints into the proposed method, including linear and quadratic constraints, equality constraints, dynamic constraints, and constraints in the form of boundaries. An important feature of the method is its computational simplicity. Complexities of the forward pass of the proposed neural network layer by linear and quadratic constraints are \(O\left( {nm} \right)\) and \(O({{n}^{2}}m)\), respectively, where n is the number of variables and m is the number of constraints. Numerical experiments illustrate the method by solving optimization and classification problems. The code implementing the method is publicly available.

中文翻译：

---

一种计算简单的新方法，用于实现具有输出硬约束的神经网络

摘要

提出了一种计算简单的新方法，对神经网络输出值施加硬凸约束。关键思想是将潜在向量映射到保证位于由一组约束定义的可行集内的点。该映射由具有输出约束的附加神经网络层实现。所提出的方法扩展到不仅对输出向量施加约束，而且对取决于输入的联合约束施加约束的情况。对输出施加约束的预测方法可以简单地在所提出的方法的框架中实现。展示了如何将不同类型的约束合并到所提出的方法中，包括线性和二次约束、等式约束、动态约束和边界形式的约束。该方法的一个重要特点是计算简单。所提出的神经网络层通过线性和二次约束的前向传递的复杂性是\(O\left( {nm} \right)\)和\(O({{n}^{2}}m)\)，分别，其中n 是变量的数量，m是约束的数量。数值实验通过解决优化和分类问题来说明该方法。实现该方法的代码是公开的。