Abstract Firms are increasingly transitioning advertising budgets to Internet display campaigns, but this transition poses new challenges. These campaigns use numerous potential metrics for success (e.g., reach or click rate), and because each website represents a separate advertising opportunity, this is also an inherently high-dimensional problem. Further, advertisers often have constraints they wish to place on their campaign, such as targeting specific sub-populations or websites. These challenges require a method flexible enough to accommodate thousands of websites, as well as numerous metrics and campaign constraints. Motivated by this application, we consider the general constrained high-dimensional problem, where the parameters satisfy linear constraints. We develop the Penalized and Constrained optimization method (PaC) to compute the solution path for high-dimensional, linearly constrained criteria. PaC is extremely general; in addition to internet advertising, we show it encompasses many other potential applications, such as portfolio estimation, monotone curve estimation, and the generalized lasso. Computing the PaC coefficient path poses technical challenges, but we develop an efficient algorithm over a grid of tuning parameters. Through extensive simulations, we show PaC performs well. Finally, we apply PaC to a proprietary dataset in an exemplar Internet advertising case study and demonstrate its superiority over existing methods in this practical setting. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

中文翻译：

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惩罚约束优化：在高维网站广告中的应用

摘要 公司越来越多地将广告预算转移到互联网展示活动，但这种转变带来了新的挑战。这些活动使用许多潜在的成功指标（例如，覆盖率或点击率），并且由于每个网站代表一个单独的广告机会，这也是一个固有的高维度问题。此外，广告商通常希望对其活动施加限制，例如针对特定的子群体或网站。这些挑战需要一种足够灵活的方法来适应数千个网站，以及众多指标和活动限制。受此应用程序的启发，我们考虑一般约束高维问题，其中参数满足线性约束。我们开发了惩罚和约束优化方法 (PaC) 来计算高维、线性约束标准的解决方案路径。PaC 非常笼统；除了互联网广告之外，我们还展示了它包含许多其他潜在应用，例如投资组合估计、单调曲线估计和广义套索。计算 PaC 系数路径带来了技术挑战，但我们在调整参数网格上开发了一种有效的算法。通过广泛的模拟，我们表明 PaC 表现良好。最后，我们将 PaC 应用于示例互联网广告案例研究中的专有数据集，并在此实际设置中证明其优于现有方法。本文的补充材料，包括对可用于复制作品的材料的标准化描述，