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：涉及在决策变量支持约束下最小化秩一凸函数的优化问题出现在各种机器学习应用中。这些问题通常通过指示变量来建模连续变量的支持。本文通过透视重构技术研究此类问题的紧凑扩展形式。与以往多数依赖支撑函数论证和析取规划技术提供凸包结论的研究不同，我们提出一种利用透视函数诱导的隐式锥结构的构造性方法。为此，我们首先建立了包含线性独立连续变量函数与二元变量组的锥混合整数集的一般凸包结论，进而证明与秩一凸函数上境图相关的集合的扩展表示在指示关系约束下自然具有此类锥表示。这使我们能够系统性地给出具有非线性可分离或不可分离目标函数、连续变量符号约束以及指示变量组合约束的集合的凸包描述的透视公式。我们在稀疏非负逻辑回归问题上验证了结果的有效性。