稀疏标准二次优化的更紧且更易处理的松弛方法及非平凡的实例生成
Tighter yet more tractable relaxations and nontrivial instance generation for sparse standard quadratic optimization
AI总结：
本文针对带稀疏约束的标准二次优化问题，提出基于现代锥优化技术的易处理凸松弛方法及其降维等价重构，并设计了避免过于简单的实例生成程序，实验证明其松弛界限高质量且求解时间对参数选择稳健。
摘要：
标准二次优化问题（StQP）可以说是所有 NP-hard 优化问题类别中最简单的一种，它由在标准单纯形（最简单的多面体/紧致可行集）上对一个二次型（最简单的非线性多项式）求极值组成。作为一个问题类别，StQP 可能是非凸的，并且具有指数级数量的低效局部解。StQP 在众多应用中出现，其中包括数学金融、机器学习（聚类）以及生物科学中的建模（例如，选择和生态学）。本文处理带有额外稀疏性或基数约束的此类 StQP，即使对于凸目标函数，这也会使问题变成 NP-hard 问题。研究这种稀疏限制下的 StQP 的一个动机是高维投资组合选择问题，其中资产过多难以处理，特别是在存在交易成本的情况下。在这里，我们依靠现代锥优化技术，为这个相关但困难的问题提出了易处理的凸松弛方法。我们提出了这些松弛方法的 novel 等价重构，具有显著的降维效果，这对于在问题规模增长时松弛方法的可处理性至关重要。此外，我们提出了一种实例生成程序，系统地避免了过于简单的实例。我们广泛的计算结果说明了在大量实例中松弛界限的高质量。此外，与精确的混合整数二次规划模型相比，松弛方法的求解时间对问题参数的选择非常稳健。特别是，降维后的公式在求解时间方面比其对应公式取得了显著的改进。
英文摘要：
The Standard Quadratic optimization Problem (StQP), arguably the simplest among all classes of NP-hard optimization problems, consists of extremizing a quadratic form (the simplest nonlinear polynomial) over the standard simplex (the simplest polytope/compact feasible set). As a problem class, StQPs may be nonconvex with an exponential number of inefficient local solutions. StQPs arise in a multitude of applications, among them mathematical finance, machine learning (clustering), and modeling in biosciences (e.g., selection and ecology). This paper deals with such StQPs under an additional sparsity or cardinality constraint, which, even for convex objectives, renders NP-hard problems. One motivation to study StQPs under such sparsity restrictions is the high-dimensional portfolio selection problem with too many assets to handle, in particular, in the presence of transaction costs. Here, relying on modern conic optimization techniques, we present tractable convex relaxations for this relevant but difficult problem. We propose novel equivalent reformulations of these relaxations with significant dimensional reduction, which is essential for the tractability of these relaxations when the problem size grows. Moreover, we propose an instance generation procedure which systematically avoids too easy instances. Our extensive computational results illustrate the high quality of the relaxation bounds in a significant number of instances. Furthermore, in contrast with exact mixed-integer quadratic programming models, the solution time of the relaxations is very robust to the choices of the problem parameters. In particular, the reduced formulations achieve significant improvements in terms of the solution time over their counterparts.
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