Optimization problems involving minimization of a rank-one convex function over constraints modeling restrictions on the support of the decision variables emerge in various machine learning applications. These problems are often modeled with indicator variables for identifying the support of the continuous variables. In this paper we investigate compact extended formulations for such problems through perspective reformulation techniques. In contrast to the majority of previous work that relies on support function arguments and disjunctive programming techniques to provide convex hull results, we propose a constructive approach that exploits a hidden conic structure induced by perspective functions. To this end, we first establish a convex hull result for a general conic mixed-binary set in which each conic constraint involves a linear function of independent continuous variables and a set of binary variables. We then demonstrate that extended representations of sets associated with epigraphs of rank-one convex functions over constraints modeling indicator relations naturally admit such a conic representation. This enables us to systematically give perspective formulations for the convex hull descriptions of these sets with nonlinear separable or non-separable objective functions, sign constraints on continuous variables, and combinatorial constraints on indicator variables. We illustrate the efficacy of our results on sparse nonnegative logistic regression problems.

翻译：涉及在决策变量支撑集约束下最小化秩一凸函数的优化问题出现在多种机器学习应用中。此类问题通常通过引入指标变量来标识连续变量的支撑集。本文利用透视重构技术研究这些问题的紧致扩展形式。与以往多数依赖支撑函数论证和析取规划技术推导凸包结论的工作不同，我们提出了一种新颖的构造方法，通过揭示透视函数所诱导的隐藏锥结构来解决问题。为此，我们首先建立了一类广义锥混合二值集的凸包结果，其中每个锥约束涉及独立连续变量的线性函数与一组二值变量。随后证明，与秩一凸函数在指标关系约束下的上境图对应的集合，其扩展表示天然具有此类锥结构。这使我们能够系统性地给出这些集合的凸包描述的透视形式，涵盖非线性可分/不可分目标函数、连续变量符号约束以及指标变量组合约束。最后通过稀疏非负逻辑回归问题验证了所得结论的有效性。