In this paper we introduce a new parameterized Quadratic Decision Rule (QDR), a generalisation of the commonly employed Affine Decision Rule (ADR), for two-stage linear adjustable robust optimization problems with ellipsoidal uncertainty and show that (affinely parameterized) linear adjustable robust optimization problems with QDRs are numerically tractable by presenting exact semi-definite program and second order cone program reformulations. Under these QDRs, we also establish that exact conic program reformulations also hold for two-stage linear ARO problems, containing also adjustable variables in their objective functions. We then show via numerical experiments on lot-sizing problems with uncertain demand that adjustable robust linear optimization problems with QDRs improve upon the ADRs in their performance both in the worst-case sense and after simulated realization of the uncertain demand relative to the true solution.

在本文中，我们介绍了一种新的参数化二次决策规则（QDR），它是常用的仿射决策规则（ADR）的推广，用于具有椭圆形不确定性的两阶段线性可调鲁棒优化问题，并证明了（仿射参数化）线性可调鲁棒通过提供精确的半定程序和二阶锥程序重新公式化，可以轻松解决QDR的优化问题。在这些QDR下，我们还确定了精确的圆锥程序重新公式化也适用于两阶段线性ARO问题，其目标函数中还包含可调整的变量。