We study the convex hull of the mixed-integer set given by a conic quadratic inequality and indicator variables. Conic quadratic terms are often used to encode uncertainties, while the indicator variables are used to model fixed costs or enforce sparsity in the solutions. We provide the convex hull description of the set under consideration when the continuous variables are unbounded. We propose valid nonlinear inequalities for the bounded case, and show that they describe the convex hull for the two-variable case. All the proposed inequalities are described in the original space of variables, but extended SOCP-representable formulations are also given. We present computational experiments demonstrating the strength of the proposed formulations.

中文翻译：

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具有指示变量的二次二次优化的强公式

我们研究了由二次二次不等式和指示变量给出的混合整数集的凸包。二次二次项通常用于编码不确定性，而指标变量用于模拟固定成本或强制解决方案中的稀疏性。当连续变量无界时，我们提供了所考虑集合的凸包描述。我们为有界情况提出了有效的非线性不等式，并表明它们描述了二变量情况的凸包。所有提议的不等式都在变量的原始空间中描述，但也给出了扩展的 SOCP 可表示公式。我们提出了计算实验，证明了所提出的公式的强度。