基于复合 Heaviside 优化的带约束分类与处理学习：一种渐进式 MIP 方法
Classification and Treatment Learning with Constraints via Composite Heaviside Optimization: a Progressive MIP Method
AI总结：
本文提出一种 Heaviside 复合优化方法及渐进式整数规划（PIP）方法，用于求解带约束的多分类与多动作处理问题，有效减轻了大型数据集下直接应用 IP 求解器的计算负担并证明了其局部最优性。
摘要：
本文提出了一种 Heaviside 复合优化方法，并给出了一种渐进式（混合）整数规划（PIP）方法，用于求解带约束的多分类和多动作处理问题。Heaviside 复合函数是 Heaviside 函数（即开区间 $( \, 0,\infty )$ 或闭区间 $[ \, 0,\infty \, )$ 的指示函数）与一个可能不可微函数的复合。在建模方面，我们展示了 Heaviside 复合优化如何为学习最优多分类和多动作处理规则提供统一公式，该规则受限于规定各种领域限制的规则依赖约束。Heaviside 复合函数具有等价的离散公式，由此产生的优化问题原则上可以通过整数规划（IP）方法求解。然而，对于具有大型数据集的约束学习问题，直接应用现成的 IP 求解器在实现全局最优性方面通常无效。为了减轻这种计算负担，我们的主要贡献是提出 PIP 方法，利用最先进的 IP 求解器对中等规模问题的有效性。我们通过与连续优化的联系提供了 PIP 方法的理论优势，并表明计算出的解对于一大类 Heaviside 复合优化问题是局部最优的。通过广泛的计算实验展示了 PIP 方法的数值性能。
英文摘要：
This paper proposes a Heaviside composite optimization approach and presents a progressive (mixed) integer programming (PIP) method for solving multi-class classification and multi-action treatment problems with constraints. A Heaviside composite function is a composite of a Heaviside function (i.e., the indicator function of either the open $( \, 0,\infty )$ or closed $[ \, 0,\infty \, )$ interval) with a possibly nondifferentiable function. Modeling-wise, we show how Heaviside composite optimization provides a unified formulation for learning the optimal multi-class classification and multi-action treatment rules, subject to rule-dependent constraints stipulating a variety of domain restrictions. A Heaviside composite function has an equivalent discrete formulation, and the resulting optimization problem can in principle be solved by integer programming (IP) methods. Nevertheless, for constrained learning problems with large data sets, a straightforward application of off-the-shelf IP solvers is usually ineffective in achieving global optimality. To alleviate such a computational burden, our major contribution is the proposal of the PIP method by leveraging the effectiveness of state-of-the-art IP solvers for problems of modest sizes. We provide the theoretical advantage of the PIP method with the connection to continuous optimization and show that the computed solution is locally optimal for a broad class of Heaviside composite optimization problems. The numerical performance of the PIP method is demonstrated by extensive computational experimentation.
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