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.

本文研究了通过透视重构技术寻找紧凑扩展表达式来解决在机器学习中出现的在约束范围内最小化排名一凸函数的优化问题，通过建立一般锥混合二进制集合的凸包结果，并展示了这种方法能够针对非线性可分离的客观函数、连续变量符号约束、指示器变量组合约束的约束关系进行透视式的解法。此方法在解决稀疏非负逻辑回归问题上表现出高效性。